THE  ELEMENTS  OF 
ELECTRICITY  AND  MAGNETISM 


MICHAEL   FARADAY  (1791-1867) 


THE  ELEMENTS  OF 

ELECTRICITY  AND  MAGNETISM 

A  TEXT-BOOK  FOR  COLLEGES 
AND  TECHNICAL  SCHOOLS 


BY 

WM.  S.  FRANKLIN  AND  BARRY   MACNUTT 

M 


Neto  ¥orfc 

THE   MACMILLAN   COMPANY 

LONDON :  MACMILLAN  &  CO.,  LTD. 

1911 

All  rights  reserved 


Engineering 
library 


COPYRIGHT  1908 
Bv  THE   MACMILLAN  COMPANY 


Set  up  and  electrotyped.     Published  July,  1908 

Reprinted  March,  1909 ;  July,  1909 

July, 1911 


PRESS  OF 

i  ERA  PRINTING  ( 
LANCASTER.  PA 


PREFACE  AND  INTRODUCTION. 

"Alles  VergSngliche  1st  nur  ein  Gleichniss." 
(Intelligibility  is  only  a  likeness. ) 

The  elementary  theory  of  electricity  and  magnetism  is  essen- 
tially an  extension  of  the  science  of  mechanics,*  and  the  purpose 
of  this  book  is  to  develop  the  science  of  electricity  and  magnetism 
from  this  point  of  view. 

The  study  of  elementary  physics,  in  one  of  its  important  phases 
is  imaginative  like  the  study  of  geometry,  its  purpose  is  to  ration- 
alize our  experience  of  physical  conditions  and  things,  and  the 
building  up  of  the  rational  structure  of  physics  should  be  the 
chief  function  of  a  text-book  for  students.  This  text  has  been 
prepared  in  accordance  with  this  idea. 

The  attempt  has  been  made  throughout  to  bring  simple  prac- 
tical applications  into  the  mind  of  the  student.  It  would  perhaps 
be  ridiculous  in  a  descriptive  treatise  on  physics  for  college  men 
to  consider  in  detail  those  things  which  are  universally  and  per- 
fectly known,  but  it  is  precisely  such  things  that  should  be  referred 
to  in  a  rational  treatise.  If  one  is  to  rationalize,  one  must  ration- 
alize about  something.  It  is  a  mistake,  however,  to  shape  science 
instruction  prematurely  to  practical  (economic)  ends,  but  such 
"  practical "  instruction  is  a  very  different  thing  from  the  rational 
study  of  the  things  of  everyday  life.  Elementary  science  instruc- 
tion must  be  made  to  touch  upon  the  things  of  everyday  life  if  it  is 
to  be  effective.  In  no  other  way  can  what  is  best  in  science  be 
realized  anew  in  each  succeeding  generation  of  men. 

*See  Art.  125  on  the  distinction  between  the  mechanical  theory  and  the  atomic 
theory. 

Special  attention  is  called  to  Art.  62  on  the  mechanical  aspect  of  Lenz's  Law  ;  to 
Chapter  VI  on  Inductance  ;  to  Art.  89  on  the  mechanical  analogue  of  the  condenser ; 
to  Arts.  106,  107  and  108  on  the  mechanical  analogies  of  electric  doubling ;  and  to 
Chapter  IX  on  the  mechanical  conceptions  of  the  electromagnetic  field  and  of  electro- 
magnetic waves. 

v 

240713 


vi  PREFACE  AND   INTRODUCTION. 

The  authors  feel  that  the  appendices  (a)  on  Terrestrial  Magnet- 
ism, (b)  on  Ship's  Magnetism  and  the  Compensation  of  the 
Compass,  (c)  on  Miscellaneous  Phenomena,  and  (d)  on  Miscel- 
laneous Practical  Applications  will  appeal  to  nearly  every  one 
who  has  occasion  to  use  this  book.  Every  student  should  know 
something  about  these  various  subjects  but  most  of  this  material 
should  be  omitted  from  a  first  systematic  study  of  the  Elements 
of  Electricity  and  Magnetism.  The  appendix  on  Ship's  Magnet- 
ism and  the  Compensation  of  the  Compass  especially  is  recom- 
mended to  those  who  wish  to  gain  a  clear  insight  into  the  physics 
of  this  subject. 

Following  the  plan  of  our  Elements  of  Mechanics,  we  wish  to 
include  an  introduction  in  this  text.  What  needs  to  be  said  in 
introduction,  however,  is  very  brief,  assuming  that  the  student 
has  read  the  introduction  to  our  Mechanics.  There  seems  to  be 
among  our  students  a  general  indifference  towards  rational 
physics  study.  What  does  this  mean?  That  all  students  are 
unworthy,  or  that  physical  science  is  at  fault  ?  Neither.  It  seems 
to  us  that  this  indifference  is  due  to  a  misunderstanding,  and  we 
believe  that  it  may  be  made  powerless  to  deter  the  student  from 
a  reasonable  expenditure  of  effort  in  the  rational  study  of  the 
physical  sciences  if  young  men  be  led  to  understand  what  kind  of 
interest  they  may  be  expected  to  have  in  such  study.  Gilbert 
Chesterton,  in  his  essays  on  Heretics,  says,  very  wisely,  that  the 
only  spiritual  or  philosophical  objection  to  steam  engines  is  not 
that  men  pay  for  them  or  work  at  them  or  make  them  very  ugly ; 
or  even  that  men  are  killed  by  them  ;  but  merely  that  men  do  not 
play  at  them.  This  is  precisely  the  objection  to  physical  science. 
Men  do  not  play  at  it,  or,  when  they  do,  it  is  play  in  the  weakest 
and  most  contemptible  sense  of  the  word.  Physical  science  in 
its  elements  is  detached  from  the  more  intensely  human  interests, 
and  the  will  alone  can  determine  its  pursuit. 

THE  AUTHORS. 

March  22,  1908. 


TABLE   OF   CONTENTS. 

CHAPTER    I. 
THE  ELECTRIC  CURRENT.     ITS  CHEMICAL  EFFECT  .         .         .         1-24 

CHAPTER   II. 
RESISTANCE  AND  ELECTROMOTIVE  FORCE.         ....       25-60 

CHAPTER   III. 
THE  MAGNETISM  OF  IRON 61-92 

CHAPTER   IV. 
MAGNETIC  EFFECT  OF  THE  ELECTRIC  CURRENT.      .         .        .     93-116 

CHAPTER  V. 
INDUCED  ELECTROMOTIVE  FORCE     ......   117-140 

CHAPTER  VI. 
ELECTRIC  MOMENTUM.     INDUCTANCE 141-159 

CHAPTER   VII. 
ELECTRIC  CHARGE.     THE  CONDENSER 160-193 

CHAPTER   VIII. 
PHENOMENA  OF  ELECTROSTATICS.     ......   194-241 

CHAPTER   IX. 
ELECTRIC  OSCILLATIONS  AND  ELECTRIC  WAVES.      .         .        .  242-275 

CHAPTER   X. 
ELECTRICAL  MEASUREMENTS 276-291 

APPENDIX   A. 

TERRESTRIAL  MAGNETISM 292-297 

vii 


viii  CONTENTS. 

APPENDIX   B. 
SHIP'S  MAGNETISM  AND  THE  COMPENSATION  OF  THE  COMPASS.  298-314 

APPENDIX   C. 
MISCELLANEOUS  PHENOMENA    .        .        •        •        •        •        •  315-322 

APPENDIX   D. 
MISCELLANEOUS  PRACTICAL  APPLICATIONS 323-341 

APPENDIX  E. 
MECHANICAL  AND  ELECTRICAL  ANALOGIES      .         .        »        .  342- 343 


CHAPTER   I. 

THE  ELECTRIC  CURRENT:  ITS  CHEMICAL  EFFECT. 

1.  The  electric  current.  —  When  a  wire  is  connected  to  the 
terminals  of  an  ordinary  battery,  certain  phenomena  are  produced 
and  an  electric  current  is  said  to  flow  through  the  wire.  A  wire 
in  which  an  electric  current  is  flowing  is  sometimes  called  an 
.electric  wire  for  brevity.  The  production  of  an  electric  current 
always  requires  a  generator  such  as  a  battery  or  a  dynamo.  The 
path  of  the  current  is  usually  a  wire  and  it  is  termed  the  electric 
circuit.  If  the  path  is  complete,  leading  out  from  the  generator 
.and  returning  to  it  without  break  or  interruption,  the  circuit  is 
said  to  be  closed ;  otherwise,  the  circuit  is  said  to  be  open.  A 
steady  electric  current  always  flows  in  a  closed  circuit,  that  is,  in 
a  circuit  which  goes  out  from  the  generator  and  returns  to  it,  and 
the  current  ceases  to  flow  when  the  circuit  is  broken. 

Certain  substances  such  as  metals  and  salt  solutions  may  form 
portions  of  an  electric  circuit.  Such  substances  are  called  electri- 
cal conductors.  Other  substances,  such  as  glass,  hard  rubber,  air, 
and  dry  wood,  cannot  form  a  portion  of  an  electric  circuit,  that 
is,  the  electric  current  cannot  flow  through  them  to  any  appreci- 
able extent.  Such  substances  are  called  insulators* 

Energy  must  be  supplied  to  an  electric  generator  (chemical 
energy  in  the  case  of  a  battery,  mechanical  energy  in  the  case  of 
.a  dynamo),  and  this  energy  reappears  in  various  parts  of  the 
electric  circuit  through  which  the  current  flows.  Thus,  energy 
reappears  as  heat  in  an  electric  lamp  and  as  mechanical  work  in 
an  electric  motor. 

The  magnetic  effect  of  the  electric  current.  —  When  an  electric 
wire  is  held  above  a  compass  and  parallel  to  the  compass  needle, 
the  compass  needle  is  deflected.  When  an  electric  wire  is 

*  All  substances  conduct  the  electric  current  more  or  less.     See  Art.  14. 
2  I 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


stretched  near  one  end  of  a  magnet,  as  shown  in  Fig.  I ,  the  wire 
is  pushed  sidewise  as  indicated  in  the  figure.     When  an  electric 


wire 


magnet 

A 

"This  wire 
is  pushed 
towards  or             — 
.away  from 
the  reader. 

S                                    N 

battery 


wire 


Fig.  1. 

current  flows  through  an  insulated  wire  which  is  wound  around 
an  iron  rod,  as  shown  in  Fig.  2,  the  iron  rod  is  magnetized,  as 
indicated  by  the  letters  WS.  These  effects  constitute  particular 

cases  of  what  may  be  called  in 
general  the  magnetic  effect  of  the 
electric  current.*  The  magnetic 
effect  of  the  electric  current,  which 
is  shown  in  its  simplest  aspect  in 
Fig  2  Fig.  I,  is  exemplified  in  a  common 

form  of  ammeter,  the  working  parts 

of  which  are  shown  in  Figs.  30  and  3^.  A  horse-shoe  magnet 
of  steel  is  provided  with  soft  iron  pole-pieces  NN  and  SS,  be- 
tween which  a  soft  iron  cylinder  C  is  rigidly  supported  by  being 
bolted  to  the  brass  strip  A.  In  the  spaces  between  the  pole- 
pieces  and  the  cylinder  C  move  the  sides  or  limbs  of  a  small 
coil  of  wire  which  is  delicately  supported  upon  a  pivot  and  which 
carries  a  pointer  which  plays  over  a  divided  scale.  Current  is 
led  into  this  movable  coil  through  the  hair-spring  at  one  end  and 
through  a  very  flexible  conductor  at  the  other  end,  and  the  side 
force  which  is  exerted  upon  the  limbs  of  the  coil  by  the  magnet 

*  Another  aspect  of  the  magnetic  effect  of  the  electric  current,  namely,  the  pro- 
duction of  current  in  a  wire  when  the  wire  is  in  motion  near  a  magnet,  is  discussed  in 
Art.  63. 


THE   ELECTRIC   CURRENT. 


poles  NN  and  55  turns  the  coil  until  this  side  force  is  balanced 
by  the  action  of  the  hair-spring. 

The  magnetic  effect  of  the  electric  current  which  is  shown  in 
its  simplest  aspect  in  Fig.   2,  is  exemplified  in  the  Morse  tele- 


Fig.  3a. 


Fig.  3b. 


graph.  A  battery  B,  Fig.  4,  is  connected  through  a  long  line 
so  as  to-  send  current  through  a  wire  which  is  wound  on  an  iron 
rod  RR  at  a  distant  station.  A  device  K,  called  a  key,  is 
arranged  for  opening  and  closing  the  circuit  through  which  the 
electric  current  flows,  and  the  rod  RR  is  magnetized  every  time 


line  wire 


ground  return 


Fig.  4. 

the  key  is  closed,  thus  causing  the  rod  RR  to  attract  a  small 
bar  of  iron  7  which  is  attached  to  a  pivoted  lever  A  ;  and  when 
the  key  K  is  opened,  the  rod  RR  loses  its  magnetism  and 
ceases  to  attract  the  iron  7.  In  this  way  the  pivoted  lever  A  is 
caused  to  move  back  and  forth  with  the  opening  and  closing  of 


4  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  key    Kt    thus  producing  any  desired  series  of  signals  at  the 
distant  station.* 

The  chemical  effect  of  the  electric  current.  —  When  a  solution 
of  a  chemical  compound  forms  apportion  of  an  electric  circuit,  the 
compound  is,  in  general,  decomposed  by  the  current.  This 
chemical  effect  of  the  electric  current  is  exemplified  in  the  prac- 
tical operation  of  electroplating.  The  essential  features  of  an 
electroplating  outfit  are  shown  in  Fig.  5.  VV 
is  a  vessel  containing,  for  example,  a  solution  of 
copper  sulphate.  The  metal  object  0  to  be 
plated  is  attached  to  one  terminal  of  a  battery 
B,  a  copper  plate  C  is  attached  to  the  other 
terminal  of  the  battery,  and  the  current  causes 
copper  to  be  deposited  upon  the  object  0. 

The  heating  effect  of  the  electric  current.  —  A 
wire,  or  any  substance  which  forms  a  portion  of  an  electric  circuit, 
has  heat  generated  in  it  by  the  current.  This  heating  effect  of 
the  electric  current  is  exemplified  in  the  ordinary  electric  lamp,  the 
carbon  filament  of  which  forms  a  portion  of  an  electric  circuit, 
and  is  heated  to  incandescence  by  the  current. 

Hydraulic  analogue  of  the  electric  current.  —  The  flow  of  an 
electric  current  through  a  circuit  of  wire  is  to  some  extent  anal- 
ogous to  the  flow  of  water  through  a  circuit  of  pipe.  The  pump 
which  propels  the  current  of  water  is  analogous  to  the  generator 
which  propels  the  electric  current,  and  the  circuit  of  pipe  which 
goes  out  from  the  pump  and  returns  to  it  is  analogous  to  the 
circuit  of  wire.  Energy  must  be  supplied  to  the  pump  to  produce 
the  flow  of  water  through  the  pipe,  and  this  energy  reappears  as 
the  heat  which  is  developed  by  the  friction  of  the  water  in  the  pipe 
or  as  the  mechanical  energy  which  is  developed  by  a  water 
motor  through  which  the  water  current  is  forced.  Similarly, 
energy  must  be  supplied  to  an  electric  generator,  and  this  energy 
reappears  in  the  electric  circuit  as  heat  or  as  the  mechanical  energy 

*  The  Morse  Telegraph  is  described  quite  fully  in  Appendix  D. 


THE    ELECTRIC    CURRENT.  5 

which  is  developed  by  an  electric  motor  through  which  the  electric 
current  is  forced. 

An  electric  current  flowing  through  a  wire  produces  an  influ- 
ence which  extends  throughout  the  region  surrounding  the  wire, 
as  is  evident  from  the  fact  that  a  compass  needle  is  deflected  when 
it  is  brought  near  an  electric  wire.  There  is,  however,  no  influ- 
ence exerted  in  the  region  surrounding  a  pipe  through  which 
water  is  flowing.  Therefore  the  hydraulic  analogue  of  the  electric 
current  is  of  no  help  in  giving  one  a  conception  of  the  magnetic 
effect  of  the  electric  current.  In  the  study  of  those  phenomena 
of  the  electric  current  which  depend  upon  its  magnetic  effect,  the 
hydraulic  analogue  must  be  used  with  caution. 

2.  The  chemical  effect  of  the  electric  current.*  —  When  a  solution 
of  a  chemical  compound  forms  a  portion  of  an  electric  circuit,  the 
compound  is,  in  general,  decomposed  by  the  current,  as  stated 
above.  Thus,  melted  salts,  and  acids  and  salts  in  solution  are 
decomposed  by  the  electric  current.  This  chemical  decomposi- 
tion is  called  electrolysis,  and  the  liquid  in  which  electrolysis  takes 
place  is  called  an  electrolyte.  Electrolysis  is  usually  carried 
out  in  a  vessel  provided  with  two  flat  plates  of  metal  or  carbon 
which  serve  to  lead  the  current  into  and  out  of  the  electrolyte. 
Such  an  arrangement  is  called  an  electrolytic  cell,  and  the  plates 
of  metal  or  carbon  are  called  the  electrodes.  The  electrode  upon 
which  the  metallic  constituent  of  the  solution  is  deposited  is  called 

*  The  chemical  effect  of  the  electric  current  is  exemplified  by  many  electrochemical 
processes  which  are  now  used  on  a  large  scale  in  various  industrial  establishments. 
See  The  Electrochemical  Manufactures  at  Niagara,  Electrochemical  Industry,  Vol.  I, 
pages  1 1-23  ;  The  Electrolytic  Refining  of  Copper,  Engineering  and  Mining 
Journal,  September  19,  1896,  and  Electrochemical  Indzistry,  Vol.  I,  page  416, 
August,  1903,  and  The  Manufacture  of  Aluminum  by  Electrolysis,  Electrochemical 
Industry,  Vol.  I,  page  158,  June,  1903. 

Perhaps  the  best  modern  treatises  on  the  phenomena  of  electrolysis  are  the 
following  : 

A  Text-book  of  Electro-chemistry  by  LeBlanc,  translated  by  W.  R.  Whitney  and 
J.  W.  Brown,  The  Macmillan  Company. 

Electro-chemistry  by  Danneel,  translated  by  Merriam,  John  Wiley  &  Sons. 

The  Theory  of  Electrolytic  Dissociation  by  H.  C.  Jones,  The  Macmillan  Company. 


6  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  cathode,  and  the  other  is  called  the  anode.  It  is  customary 
to  speak  of  the  current  as  flowing  into  an  electrolytic  cell  at  the 
anode  and  out  of  the  cell  at  the  cathode,  that  is,  the  electric  cur- 
rent is  considered  to  flow  in  the  c&rection  in  which  the  metallic  con- 
stituent of  the  solution  is  carried  in  an  electrolytic  cell. 

Consider  a  solution  of  hydrobromic  acid  (HBr).  When  an 
electric  current  is  passed  through  this  solution,  hydrogen  (H) 
is  liberated  at  the  cathode  and  bromine  (Br)  is  liberated  at  the 
anode.  In  general,  the  molecule  of  any  dissolved  salt  or  acid  is 
separated  into  two  parts  by  electrolysis ;  one  part  is  liberated  at 
the  cathode  and  is  called  the  cathion,  and  the  other  part  is  liber- 
ated at  the  anode  and  is  called  the  anion.  Thus,  hydrogen  (H) 
is  the  cathion  and  bromine  (Br)  is  the  anion  of  hydrobromic 
acid.  In  all  metallic  salts  the  metal  constitutes  the  cathion  and 
the  acid  radical  or  halogen  constitutes  the  anion.  In  acids  the 
hydrogen  constitutes  the  cathion  and  the  acid  radical  or  halogen 
constitutes  the  anion.  Thus,  the  cathion  of  copper  sulphate 
(CuSO4)  is  copper  (Cu),  and  the  anion  is  the  acid  radical  (SO4). 

In  many  cases  the  cathion  and  anion  are  not  actually  liberated 
at  the  electrodes  because  of  what  are  called  secondary  reactions. 
Thus,  in  the  electrolysis  of  an  aqueous  solution  of  sodium  chloride 
(NaCl),  the  cathion  (Na),  when  it  is  liberated  at  the  cathode, 
immediately  reacts  upon  the  water,  forming  NaOH  and  free 
hydrogen  ;  in  the  electrolysis  of  copper  sulphate  between  copper 
electrodes,  the  anion  (SO4)  combines  with  the  copper  of  the 
anode  forming  fresh  CuSO4  which  goes  into  solution  or  is 
deposited  as  crystals  on  the  anode  if  the  solution  is  saturated ;  in 
the  electrolysis  of  H2SO4  between  inert  electrodes  such  as  car- 
bon or  platinum,  the  hydrogen  is  liberated  at  the  cathode  as  a 
gas,  and  the  anion  (SO4)  reacts  on  the  water  according  to  the 
formula  SO4  +  H2O  =  H2SO4  +  O  and  the  free  oxygen  escapes 
as  gas.  The  reason  for  taking  the  unfamiliar  substance  hydro- 
bromic acid  in  the  above  example  is  that  in  the  electrolysis  of 
hydrobromic  acid  there  are  no  secondary  reactions  at  the 
electrodes. 


THE   ELECTRIC   CURRENT.  7 

The  chemical  action  which  is  caused  by  the  flow  of  current 
through  an  electrolytic  cell  is  confined  wholly  to  the  immediate 
neighborhood  of  the  electrodes.  This  is  exemplified  by  passing 
an  electric  current  through  a  solution  of  lead  nitrate  between  lead 
electrodes  in  a  narrow  glass  vessel  which  can  be  placed  before  the 
lantern  and  projected  on  the  screen.  The  lead  is  deposited  upon 
the  cathode  in  beautiful  feather-like  crystals,  and  the  solution  in 
the  immediate  neighborhood  of  the  cathode  becomes  less  dense 
as  the  lead  is  deposited  out  of  it  upon  the  cathode  as  may  be  seen 
by  the  upward  streaming  of  the  solution  near  the  surface  of  the 
cathode.  On  the  other  hand,  the  solution  near  the  anode  is 
increased  in  density  by  the  dissolving  of  the  lead  of  the  anode  by 
the  NO3  which  is  liberated  there  by  the  current,  as  may  be  seen 
by  the  downward  streaming  of  the  solution  in  the  neighborhood 
of  the  anode.  The  solution  remains  entirely  unchanged  through- 
out the  region  between  the  electrodes.* 

The  dissolving  of  the  metal  of  the  anode  may  be  observed 
directly  by  reversing  the  current,  thus  causing  the  feather-like 
crystals  of  lead  which  have  already  been  deposited  upon  one  of 
the  lead  electrodes  to  become  the  anode.  Under  these  conditions 
the  crystals  are  seen  to  dissolve  rapidly. 

3.  Measurement  of  current  by  its  chemical  effect.  Definition  of 
the  ampere.  —  The  electric  current  in  a  wire  may  be  measured  in 
terms  of  its  magnetic  effect,  or  in  terms  of  its  heating  effect,  or  in 
terms  of  its  chemical  effect.  Thus,  it  would  be  permissible  to 
think  of  one  current  as  being  twice  as  strong  as  another  if  it  would 
produce  twice  as  much  heat  per  second  as  the  other  current  when 
it  is  allowed  to  flow  through  a  given  wire  ;  f  but  the  magnetic 
effect  has  been  adopted  as  the  basis  of  current  measurement  as 
fully  explained  in  Chapter  IV.  The  measurement  of  current  by 
its  chemical  effect,  however,  is  consistent  with  the  fundamental 
measurement  by  magnetic  effect,  and  therefore,  we  may  for  the 

*  Except  for  a  slight  rise  of  temperature  due  to  the  heating  effect  of  the  current, 
f  A  definition  of  current  strength  on  this  basis  would  lead  to  a  more  complicated 
scheme  of  electrical  theory  than  that  at  present  in  vogue. 


8  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

present  define  the  strength  of  an  electric  current  as  proportional 
to  the  amount  of  a  given  metal  deposited  by  the  current  per  second 
in  an  electrolytic  cell. 

The  international  standard  ampere  is  defined  *  as  that  strength 
of  current  which  will  deposit  o.ooi  1 18  gram  of  silver  per  second 
from  an  aqueous  solution  of  pure  silver  nitrate.  Another  unit  of 
current,  the  abampere  or  c.g.s.  unit,  is  defined  in  Art.  52. 

The  coulombmeter.  —  An  electrolytic  cell  arranged  for  the  meas- 
urement of  current  by  weighing  the  amount  of  metal  deposited  by 
the  current  in  a  given  time  is  called  a  coulombmeter.f  Thus, 
the  copper  coulombmeter  consists  of  a  glass  vessel  containing  an 
aqueous  solution  of  copper  sulphate  and  having  sheet-copper 
electrodes.  The  cathode,  or  gain-plate,  is  weighed  at  the  begin- 
ning and  again  at  the  end  of  the  run,  and  the  strength  of  the  cur- 
rent is  calculated  by  dividing  the  observed  amount  of  deposited 
copper  by  the  amount  of  copper  that  would  be  deposited  in  the 
same  time  by  one  ampere. 

Current  density  at  an  electrode.  —  The  quotient  of  the  current 
flowing  through  an  electrolytic  cell  divided  by  the  active  area  of 
one  of  the  electrodes  is  called  the  current  density  at  that  electrode. 
The  physical  character  of  the  metal  which  is  deposited  by  an 
electric  current  depends  very  greatly  upon  the  current  density  at 
the  electrode  upon  which  the  metal  is  deposited.  Thus,  metallic 
copper  is  deposited  from  a  solution  of  copper  sulphate  as  a 
smooth,  solid  layer  if  the  current  density  does  not  exceed  0.02 
ampere  per  square  centimeter,  whereas  the  deposit  becomes  very 
rough  with  projecting  crystals  of  the  metal  if  the  current  density 

*  In  accordance  with  the  recommendations  of  the  International  Electrical  Congress 
which  met  at  Chicago  in  1893.  The  fundamental  definition  of  the  ampere  is  based 
upon  the  magnetic  effect  of  the  electric  current  as  explained  in  Art.  52.  The  value 
of  a  current  in  amperes  (as  defined  by  the  magnetic  effect)  may  be  determined  from 
purely  mechanical  measurements  as  explained  in  Art.  59.  In  this  way  the  amount  of 
silver  deposited  in  one  second  by  one  ampere  may  be  determined.  This  determination 
has  been  made  a  number  of  times  with  great  care,  the  latest  determination  being  that 
of  H.  S.  Carhart  and  G.  W.  Patterson.  See  Journal  of  'the  Institution  of  Electrical 
Engineers,  Vol.  34,  pages  185-189,  February,  1905. 

f  Sometimes  called  a  voltameter. 


THE   ELECTRIC   CURRENT.  9 

is  greatly  in  excess  of  this.  The  character  of  the  chemical  action 
which  takes  place  at  an  electrode  also  depends  upon  the  current 
density.  Thus,  copper  alone  is  deposited  upon  a  cathode  from  a 
mixed  solution  of  zinc  and  copper  sulphates  if  the  current  density 
is  very  small,  whereas  a  mixture  of  copper  and  zinc  is  deposited 
upon  the  cathode  if  the  current  density  is  excessive.* 

4.  Faraday's  lawsf  of  electrolysis.  First  law.  —  The  amount 
of  a  given  metal  which  is  deposited  electrolytically  is  propor- 
tional to  the  strength  J  of  the  current  and  to  the  time,  that  is, 

M=klt  (i) 

in  which  M  is  the  amount  of  metal  in  grams  deposited  in  / 
seconds  by  a  current  of  /  amperes,  and  k  is  a  constant  for  a 
given  metal.  This  constant  k  is  called  the  electrochemical  equiva- 
lent of  the  given  metal.  Electrochemical  equivalents  are  ordinarily 
specified  in  grams  of  metal  deposited  per  ampere  of  current  per 
second. 

Second  law.  —  The  electrochemical  equivalents  of  elements 
which  can  form  the  ions  of  an  electrolyte,  are  proportional  to  the 
quotients  of  their  atomic  weights  divided  by  their  valencies.  A 
metal  which  has  two  valencies  has  two  values  for  its  electrochem- 
ical equivalent.  Thus  one  and  one  half  times  as  much  iron  is 

*  The  deposition  of  one  metal  instead  of  several  from  solutions  of  mixed  salts 
depends  more  distinctly  upon  the  electromotive -force  drop  between  the  electrode  and 
the  solution  (electrode  polarization)  than  upon  current  density.  See  Art.  22. 

f  The  laws  of  physics  are  the  experimental  facts  upon  which  the  science  is  based. 
Thus  Faraday' s  laws  of  electrolysis  are  the  result  of  experiment,  pure  and  simple  ; 
Boyle's  and  Gay  Lussac'  s  Laws  concerning  the  expansion  of  gases  are  experimental  facts; 
Newton's  Laws  of  Motion  are  experimental  facts  ;  Newton's  Law  of  Gravitation  is  an 
experimental  fact  ;  and  so  on.  In  nearly  every  case  the  so-called  laws  of  physics  are 
only  approximately  true.  Thus,  the  product  of  the  volume  and  pressure  of  a  given 
amount  of  gas  at  constant  temperature  is  not  strictly  constant  (Boyle's  Law)  ;  the 
amount  of  metal  deposited  by  an  electric  current  deviates  in  many  cases  from  an  exact 
proportional  relationship  with  the  current  (see  Practical  Physics,  Franklin,  Craw- 
ford and  MacNutt,  Vol.  II,  page  136). 

J  In  Faraday's  experiments,  which  led  to  the  formulation  of  this  general  law,  the 
electric  current  was  measured  by  a  galvanometer,  that  is,  the  electric  current  was 
measured  in  terms  of  its  magnetic  effect. 


10          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

deposited  from  a  solution  of  a  ferrous  salt  as  from  a  solution  of  a 
ferric  salt,  provided  that  the  deposition  is  not  complicated  by 
secondary  reactions  which  cause  the  deposit  to  be  redissolved 
chemically. 

5.  The  dissociation  theory  of  electrolysis.  —  The  molecules  of 
an  electrolytic  salt  or  acid  when  in  solution,  or  when  melted,  are 
thought  to  be  more  or  less  dissociated  into  what  are  called  ions. 
For  example,  the  molecules  of  copper  sulphate  (CuSO4)  in  a 
dilute  aqueous  solution  are  all  dissociated  into  Cu  (cathions)  and 
SO4  (anions) ;  the  molecules  of  sodium  chloride  (NaCl)  in  a 
dilute  aqueous  solution  are  all  dissociated  into  Na  (cathions) 
and  Cl  (anions).  These  ions  are  supposed  to  be  electrically 
charged*  and  to  wander  about  through  the  solution.  When  an 
electric  current  flows  through  the  electrolyte,  the  positively 
charged  ions  (cathions)  move  towards  the  cathode  where  they 
part  with  their  positive  charges  and  are  deposited  as  hydrogen 
or  metal,  as  the  case  may  be,  and  the  negatively  charged  ions 
(anions)  move  towards  the  anode  where  they  part  with  their 
negative  charges.  This  movement  of  positively  and  negatively 
charged  ions  constitutes  the  electric  current  in  the  electrolyte. 

Conception  of  Faraday1  s  first  law.  — All  of  the  ions  of  a  given 
substance  have  the  same  electric  charge  so  that  the  strength  of 
the  current  is  proportional  to  the  number  of  ions  deposited  per 
second  on  one  of  the  electrodes. 

Conception  of  Faraday's  second  law.  —  All  monovalent  ions 
carry  the  same  amount  of  charge,  the  charge  on  a  monovalent 
cathion  being  positive  and  the  charge  on  a  monovalent  anion 
being  negative.  For  example,  the  cathions  in  the  following 
series  of  chlorides  are  all  monovalent,  hydrochloric  acid  (HC1), 
potassium  chloride  (KC1),  sodium  chloride  (NaCl),  and  cuprous 
chloride  (Cud),  and  the  same  number  of  atoms  of  H,  K,  Na,  and 
Cu  are  deposited  from  solutions  of  these  chlorides  in  a  given 
time  by  a  given  current,  so  that  the  electrochemical  equivalents 

*  See  Art.  85  for  definition  of  electric  charge. 


THE   ELECTRIC   CURRENT.  II 

of  these  monovalent  metals  are  directly  proportional  to  their 
atomic  weights. 

The  charge  on  an  ion  is  proportional  to  its  valency.  Thus,  the 
copper  ion  in  a  solution  of  cupric  chloride  (CuCl2)  has  twice  as 
much  charge  as  the  copper  ion  in  a  solution  of  cuprous  chloride 
(CuCl),  so  that  half  as  many  cupric  ions  as  cuprous  ions  are  de- 
posited by  a  given  current  in  a  given  time.  In  general,  if  n  is 
the  number  of  monovalent  ions  deposited  in  one  second  by  one 
ampere,  then  nJ2  is  the  number  of  bivalent  ions  deposited  in 
the  same  time  by  the  same  current,  nj  $  is  the  number  of  trivalent 
ions  deposited  in  the  same  time  by  the  same  current,  and  so  on. 

Consider  a  series  of  chlorides  of  metals  of  different  valencies, 
for  example,  sodium  chloride  (NaCl),  cupric  chloride  (CuCl2), 
ferric  chloride  (FeCl3),  and  stannic  chloride  (SnCl4).  Reduc- 
ing these  all  to  a  given  amount,  say  n  atoms,  of  chlorine,  we 
would  have  n  atoms  of  sodium  (Na),  nJ2  atoms  of  copper 
(Cu),  «/3  atoms  of  iron  (Fe),  and  n/4  atoms  of  tin  (Sn);  so 
that,  during  the  liberation  of  n  atoms  of  chlorine,  we  would 
have  a  deposit  of  n  atoms  of  sodium  (Na),  nJ2  atoms  of 
copper  (Cu),  nj $  atoms  of  iron  (Fe),  and  n/4  atoms  of  tin 
(Sn).  Therefore,  the  weights  of  these  various  metallic  deposits 
would  be  proportional  to  their  atomic  weights  divided  by  their 
respective  valencies. 

Let  us  represent  each  unit  of  charge  by  a  plus  or  minus  sign. 
Then  the  single,  double,  triple  and  quadruple  charges  on  the  ions 
of  sodium,  copper,  iron  and  tin  may  be  represented  as  follows : 
Na  ,  Cu+,  Fef  and  Sn|,  and  the  single  and  double  charges  upon 
the  monovalent  and  bivalent  anions  of  chlorine  and  SO4  may  be 
represented  as  follows :  —  Cl  and  H  SO4.  The  present  hypothe- 
sis concerning  chemical  affinity  is  that  it  is  due  to  the  attraction 
ot  the  opposite  charges  on  the  two  constituents  of  the  molecule. 
Thus,  sodium  and  chlorine  are  held  together  in  the  molecule  of 
sodium  chloride  by  the  attraction  of  the  positive  charge  on  the 
sodium  for  the  negative  charge  on  the  chlorine,  as  may  be  repre- 
sented thus  :  Na  +  —  Cl. 


12  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

One  of  the  greatest  difficulties  in  the  dissociation  theory  of 
electrolysis  is  to  account  for  the  breaking  up  of  such  a  molecule 
as  sodium  chloride,  which  is  ordinarily  very  stable,  into  its  ions. 
The  strength  of  the  dissociation,,  theory,  however,  lies  in  the 
extent  to  which  it  correlates  a  wide  range  of  experimental  fact, 
and  in  this  respect  the  dissociation  theory  is  incomparably  more 
useful  than  any  other  theory  that  has  been  hitherto  proposed.* 

6.  The  voltaic  cell.  —  The  chemical  action  that  is  caused  by 
the  flow  of  current  through  an  electrolytic  cell  is  usually  forced, 
that  is,  work  has  to  be  done  to  bring  the  chemical  action  about 
or,  in  other  words,  an  electric  generator  such  as  a  dynamo  or  a 
battery  must  be  used  to  push  the  current  through  the  electrolytic 
cell.  When,  however,  secondary  chemical  actions  take  place  at 
one  or  both  electrodes,  it  frequently  happens  that  the  total  chem- 
ical action  that  is  brought  about  by  the  flow  of  current  through 
an  electrolytic  cell  is  a  source  of  energy.  In  such  a  case  the 
electrolytic  cell  itself  can  maintain  its  own  current  through  the 
electrolyte  from  electrode  to  electrode  and  through  an  outside 
circuit  of  wire  which  connects  the  electrodes.  Such  an  electro- 
lytic cell  is  called  a  voltaic  cell. 

Example.  —  When  a  strip  of  clean  zinc  and  a  strip  of  copper  or 
carbon  are  dipped  into  dilute  sulphuric  acid,  no  appreciable  chem- 
ical action  takes  place.  When  the  plates  are  connected  together 
by  a  wire,  a  current  immediately  starts  to  flow  through  the  circuit, 
leaving  the  cell  at  the  copper  or  carbon  electrode  (the  cathode)  and 

*  Several  simple  applications  of  the  dissociation  theory  to  the  interpretation  of  ex- 
perimental results  are  given  in  Practical  Physics,  Franklin,  Crawford  and  Mac- 
Nutt,  Vol.  II,  page  108,  page  144  and  pages  146  and  147.  A  splendid  example  of 
the  application  of  the  dissociation  theory  to  the  rationalization  of  a  very  complicated 
experimental  result  is  given  by  E.  C.  Franklin  and  H.  D.  Gibbs  in  \.\\.&  Journal  of 
the  American  Chemical  Society,  Vol.  29,  pages  1389-1396,  October,  1907. 

Any  student  who  wishes  to  become  acquainted  with  the  facts  of  electrolysis  must 
familiarize  himself  with  the  details  of  the  dissociation  theory,  and,  since  no  other 
theory  has  ever  been  proposed  which  is  to  be  compared  in  effectiveness  with  the  dis- 
sociation theory,  the  student's  efforts  should  be  directed  first  of  all  to  a  thorough 
understanding  of  the  theory.  After  he  has  mastered  the  theory  its  imperfections  may 
properly  be  pointed  out. 


THE   ELECTRIC    CURRENT.  13 

entering  the  cell  at  the  zinc  electrode  (the  anode).  This  current 
decomposes  the  sulphuric  acid  (H2SO4),  the  hydrogen  is  liber- 
ated at  the  copper  or  carbon  cathode  and  escapes  from  the  cell 
as  a  gas,  and  the  sulphuric  acid  radical  (SO4),  which  is  set  free 
at  the  zinc  anode,  combines  with  the  zinc  and  forms  zinc  sulphate 
(ZnSO4)  which  goes  into  solution.  The  combination  of  Zn  and 
SO4  develops  more  energy  than  is  required  for  the  decomposi- 
tion of  the  H2SO4  so  that  the  chemical  action  as  a  whole  is  a 
source  of  energy. 

The  available  energy  of  the  reaction  above  described  may  be 
greatly  increased  by  providing  an  oxidizing  agent  in  the  neighbor- 
hood of  the  cathode  so  that  the  hydrogen  may  be  oxidized  and 
form  water  (H2O)  at  the  moment  of  its  liberation  by  the  current. 
The  energy  of  this  oxidation  is  then  added  to  the  available  energy 
of  the  total  chemical  action  in  the  cell.* 

7.  Examples  of  voltaic  cells.  The  ordinary  "  dry  cell."  —  One 
of  the  most  familiar  types  of  voltaic  cell  is  the  cell  in  which 
a  plate  of  zinc  and  a  plate  of  carbon  are  immersed  in  a  solution 
of  ammonium  chloride  (NH4C1)  with  a  mass  of  powdered  black 
oxide  of  manganese  (MnO2)  packed  around  the  carbon  electrode. 
When  this  cell  delivers  current,  the  NH4C1  is  decomposed,  and 
chlorine  is  liberated  at  the  zinc  plate  where  it  combines  with  the 
zinc  to  form  zinc  chloride.  As  the  NH4  ions  are  liberated  at 
the  carbon  electrode  they  break  up  into  ammonia  and  hydrogen 
(NH4  =  NH3  -f  H),  the  ammonia  goes  into  solution  and  the 
hydrogen  is  oxidized  at  the  expense  of  the  oxygen  in  the  black 
oxide  of  manganese,  forming  water.  The  free  ammonia  in  this 
type  of  cell  may  be  detected  by  the  odor  after  the  cell  has  been 
delivering  current  for  some  time. 

This  type  of  cell  is  exemplified  by  a  great  variety  of  commer- 
cial forms  of  which  the  ordinary  "dry  cell"  is  the  most  familiar. 
In  this  cell  the  electrolyte  is  soaked  up  in  a  porous  material  such 

*The  student  is  referred  to  Professor  H.  S.  Carhart's  Primary  Batteries,  pub- 
lished by  Allyn  &  Bacon,  Boston,  Mass.,  for  full  information  on  primary  batteries 
(voltaic  cells)  and  primary  battery  tests. 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


as  saw-dust,  the  containing  vessel  is  made  of  zinc  and  serves  as 
the  zinc  electrode,  and  the  cell  is  hermetically  sealed  so  as  to  pre- 
vent evaporation. 

The  ordinary  gravity  .cell,  which  is  shown  in  Fig.  6,  consists 
of  a  copper  electrode  in  the  bottom  of  a  jar  surrounded  by  a  solu- 
tion of  copper  sulphate,  and  a  zinc  electrode  in  the  top  of  the  jar 
surrounded  by  a  solution  of  zinc  sulphate.  The  light  zinc  sul- 
phate solution  floats  on  the  heavy  copper  sulphate  solution. 
When  this  cell  delivers  current,  SO4  is  liberated  at  the  zinc 
electrode  where  it  combines  with  the  zinc  forming  additional  zinc 


Fig.  6. 


Fig.  7. 


sulphate,  and  metallic  copper  is  deposited  upon  the  copper  elec- 
trode at  the  bottom  of  the  cell.  When  this  cell  is  in  use,  the 
copper  sulphate  must  be  replenished  occasionally  by  dropping 
fresh  crystals  of  the  salt  into  the  cell,  and  a  portion  of  the  zinc 
sulphate  solution  must  be  occasionally  drawn  off  and  replaced  by 
water. 

The  chromic  acid  cell  consists  of  a  plate  of  amalgamated  zinc 
and  a  plate  of  carbon  dipping  into  a  solution  of  a  mixture  of 
chromic  acid  (H2Cr2O7)  and  sulphuric  acid  (H2SO4).  When 
this  cell  delivers  current,  the  flow  of  the  current  through  the  cell 
decomposes  the  H2SO4.  The  acid  radical  SO4  is  liberated  at 


THE   ELECTRIC   CURRENT. 


the  zinc  electrode  where  it  combines  with  the  zinc  forming  zinc  sul- 
phate, and  the  hydrogen  is  liberated  at  the  carbon  electrode  where 
it  is  oxidized  at  the  expense  of  the  oxygen  in  the  chromic  acid. 

In  the  chromic  acid  cell,  the  zinc  wastes  away  rapidly  even 
when  the  cell  is  not  delivering  current,  and  it  is  therefore  desirable 
to  lift  the  zinc  out  of  the  solution  when  the  cell  is  not  in  use. 
Figure  7  shows  a  chromic  acid  cell  arranged  so  that  the  zinc 
electrode  may  be  conveniently  lifted  out  of  the  solution.  In  this 
figure  the  cell  is  shown  with  a  zinc  electrode  placed  between 
two  carbon  plates.  The  two  carbon  plates  are  connected  together 
and  constitute  one  electrode. 

The  Edison-LaLande  cell  consists  of  a  zinc  plate  and  a  compact 
block  of  copper  oxide  (CuO)  immersed  in  a  strong  solution  of 
caustic  potash  (KOH).  The  cell  shown  in  Fig.  8  has  two  zinc 


SolutIonHfe3= 


Pt.w! 


Paste 


Hg, 


Fig.  8. 


Fie.  9. 


plates  on  opposite  sides  of  the  copper  oxide  plate.  These 
two  zinc  plates  are  connected  together  and  constitute  a  single 
electrode.  When  this  cell  delivers  current,  the  KOH  is  decom- 
posed, potassium  ions  are  liberated  at  the  copper  oxide  plate,  the 
copper  oxide  is  reduced  to  metallic  copper,  and  the  potassium  is 
oxidized  and  goes  into  solution  as  KOH.  At  the  same  time 
hydroxyl  ions  (OH)  are  liberated  at  the  zinc  electrode  where 
they  break  up  into  free  oxygen  and  water  (2OH  =  O  +  H2O), 
the  free  oxygen  combines  with  the  zinc  forming  zinc  oxide,  and 


16  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

this  zinc  oxide  combines  with  the  caustic  potash  in  the  solution 
forming  potassium  zincate  (K2ZnO2). 

The  Clark  standard  cell  is  arranged  as  shown  in  Fig.  9.  One 
electrode  is  pure  mercury  and  the  other  electrode  is  pure  zinc. 
When  this  cell  delivers  current,  the  ZnSO4  in  solution  is  decom- 
posed, SO4  is  liberated  at  the  surface  of  the  zinc  where  it  com- 
bines with  the  zinc  forming  ZnSCX,  and  at  the  same  time  Zn  is 
liberated  at  the  surface  of  the  mercury  electrode  where  it  is  acted 
upon  by  the  mercurous  sulphate  Hg9SO4  according  to  the 
equation 

Zn  +  Hg2S04  =  Hg2  +  ZnS04 

This  cell  is  remarkable  for  the  constancy  of  its  electromotive  force 
and  it  is  used  as  a  standard  of  electromotive  force,  as  explained 
in  Chapter  X. 

8,  Voltaic  action  and  local  action.  —  Two  kinds  of  chemical 
action  are  to  be  distinguished  in  a  voltaic  cell,  namely,  (a)  the 
chemical  action  which  depends  upon  the  flow  of  current  and  does 
no';  exist  when  there  is  no  current  and  (b)  the  chemical  action 
which  is  independent  of  the  flow  of  current  and  which  takes 
place  whether  the  current  is  flowing  or  not. 

The  chemical  action  which  depends  on  the  current  is  propor- 
tional to  the  current,  it  is  essential  to  the  operation  of  the  voltaic 
cell  as  a  generator  of  current,  its  energy  is  available  for  the  mainte- 
nance of  the  current,  and  it  is  called  voltaic  action. 

The  chemical  action  in  a  voltaic  cell  which  is  independent  of 
the  flow  of  current  does  not  help  in  any  way  to  maintain  the  cur- 
rent, it  represents  absolute  waste  of  materials,  and  it  is  called  local 
action.  Local  action  takes  place  more  or  less  in  every  type  of 
voltaic  cell  and  it  is  especially  marked  in  the  chromic  acid  cell 
above  described.  It  may  be  reduced  to  a  minimum  in  a  given 
type  of  voltaic  cell  by  coating  the  zinc  with  a  thin  layer  of  metallic 
mercury  (amalgamation). 

The  term,  local  action,  originated  in  the  following  considerations  :  When  a  strip  of 
clean  zinc  is  immersed  in  sulphuric  acid,  no  perceptible  chemical  action  takes  place. 


THE   ELECTRIC   CURRENT.  I/ 

If  the  zinc  is  connected  to  a  carbon  or  copper  electrode,  however,  or  if  a  piece  of  car. 
bon  or  copper  touches  the  zinc  plate  in  the  solution,  chemical  action  begins  at  once, 
current  flows  through  the  electrolyte  from  the  zinc  to  carbon  or  copper  and  back 
through  the  metallic  connection  to  the  zinc,  the  sulphuric  acid  is  decomposed,  hydro- 
gen is  liberated  at  the  carbon  or  copper  electrode,  and  SO4  is  liberated  at  the  zinc 
electrode  where  it  combines  with  the  zinc  forming  zinc  sulphate.  When  a  plate  of 
impure  zinc  is  immersed  in  dilute  sulphuric  acid,  the  insoluble  impurities  are  left  in 
the  form  of  fine  particles  clinging  to  the  surface  of  the  zinc  after  the  zinc  is  partly  dis- 
solved, and  these  fine  particles  play  the  part  of  carbon  or  copper  cathodes,  current  flows 
through  the  acid  from  the  zinc  to  each  particle  and  back  to  the  zinc  through  the  point 
of  attachment  of  the  particle  with  the  zinc  plate,  as  indicated  in  Fig.  loa,  the  acid  is 
decomposed,  hydrogen  is  liberated  at  the  surface  of  each  par- 
ticle, and  SO4  is  liberated  at  the  surface  of  the  zinc  plate 
where  it  combines  with  the  zinc  forming  zinc  sulphate.  The 
rapid  dissolving  of  impure  zinc  in  sulphuric  acid  is  no  £  zinc 


doubt  due  to  the  flow  of  electric  currents  through  the  min-     .,  H       * 

ute  "local  circuits"  as  here  described,  and  this  rapid  dis-  • ~ 

solving  of  impure  zinc  is  therefore  called  local  action.  . 

The  covering  of  the  zinc  plate  with  a  thin  layer  of  me-  acid 

tallic  mercury  tends  to  produce  a  clean  metallic  surface 
which  is  free  from  adhering  particles  of  the  impurity  which 
is  left  as  the  zinc  wastes  away,  and  the  above  described  ac-  pj  10a 

tion  does  not  take  place.     It  is  probable  that  in  some  cases 

chemical  action  (local  action)  takes  place  irrespective  of  the  flow  of  electric  currents 
in  local  circuits  as  above  described.  This  seems  to  be  the  case,  for  example,  in  the 
chromic  acid  cell,  for,  as  a  matter  of  fact,  more  than  three  fourths  of  the  zinc  in  such 
a  cell  is  consumed  independently  of  voltaic  action,  even  when  the  zinc  is  thoroughly 
amalgamated  so  as  to  present  a  clean  bright  surface,  but  in  the  chromic  acid  cell  the 
local  action  is  very  much  less  when  the  zinc  is  amalgamated  than  it  is  when  the  zinc 
is  not  amalgamated. 

An  essential  feature  of  voltaic  action  is  that  it  is  reversed  if  a 
current  is  forced  backwards  through  a  voltaic  cell  by  an  outside 
agent,  provided  that  no  material  that  has  played  a  part  in  the 
previous  voltaic  action  has  been  allowed  to  escape  from  the 
cell.  Thus  in  the  operation  of  the  simple  voltaic  cell  consisting 
of  a  zinc  anode  and  a  carbon  cathode  in  dilute  sulphuric  acid,  the 
H2SO4  is  decomposed,  ZnSO4  is  formed  at  the  anode,  and  hydro- 
gen is  liberated  at  the  cathode.  If  the  current  is  reversed  so  that 
the  carbon  plate  becomes  the  anode,  and  the  zinc  plate  the 
cathode,  then  the  ZnSO4,  previously  formed,  will  be  decomposed, 
metallic  zinc  will  be  deposited  upon  the  zinc  cathode,  and  SO4 
will  be  liberated  at  the  carbon  anode  where  it  will  combine  with 
3 


1 8  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  trace  of  hydrogen  that  is  clinging  to  the  carbon  plate  and 
form  H2SO4.  In  this  cell,  however,  the  greater  part  of  the 
liberated  hydrogen  has,  of  course,  escaped,  and  the  reversed 
chemical  action  due  "to  a  reversed  current  cannot  long  continue. 
Local  action,  on  the  other  hand,  being  independent  of  current,  is 
not  affected  by  a  reversal  of  the  current. 

9.  The  storage  cell.*  —  A  voltaic  cell  which  is  free  from  local 
action  and  in  which  all  of  the  materials  which  take  part  in  the 
voltaic  action  are  conserved  in  the  cell,  may  be  regenerated  after 
use  by  sending  through  it  a  reversed  current.  This  regeneration 
is  due  to  the  reversed  chemical  action  that  is  produced  by  the 
reversed  current  as  explained  in  the  previous  article.  A  voltaic 
cell  that  is  adapted  to  be  thus  regenerated,  that  is,  a  voltaic  cell 
in  which  there  is  no  local  action  and  in  which  all  of  the  materials 
which  take  part  in  the  voltaic  action  are  conserved  in  the  cell,  is 
called  a  storage  cell.  The  process  of  regeneration  is  called 
charging,  and  the  use  of  the  cell  as  an  electric  generator  is  called 
discharging.  A  storage  cell  always  requires  more  energy  to 
charge  it  than  is  delivered  by  the  cell  during  the  discharge. 

The  lead  storage  cell.  —  The  voltaic  cell  which,  up  to  the 
present  time,  has  been  found  to  be  most  satisfactory  when  used 
as  a  storage  cell,  is  a  voltaic  cell  having  a  cathode  of  lead  peroxide 
(PbO2),  an  anode  of  spongy  metallic  lead,  and  an  electrolyte  of 
dilute  sulphuric  acid.  The  lead  peroxide  and  the  spongy  metallic 
lead  are  both  converted  into  insoluble  lead  sulphate  (PbSOJ 
when  the  cell  is  discharged.  When  this  cell  is  charged,  the 
lead  sulphate  is  converted  back  into  lead  peroxide  and  spongy 
lead  respectively.  The  lead  peroxide  and  the  spongy  lead  are 

*The  description  here  given  of  the  action  of  the  lead  storage  cell  is  a  simple 
working  theory  of  the  cell.  The  actions  as  described  do,  no  doubt,  take  place,  but 
they  are  complicated  by  more  complex  actions  such  as  the  formation  of  persulphates 
at  the  anode  and  of  subsulphates  at  the  cathode.  See  The  Theory  of  the  Lead 
Accumulator •,  by  Friedrich  Dolezalek  (English  translation  by  C.  L.  von  Ende, 
published  by  John  Wiley  &  Sons).  A  good  engineering  treatise  on  the  storage 
battery  is  Storage  Battery  Engineering  by  Lamar  Lyndon  (McGraw  Publishing 
Company). 


THE   ELECTRIC   CURRENT. 


Fig.  lOb. 


called  the  active  materials  of  the  cell.     These  active  materials 
are  mechanically  weak  and  porous  and  they  are  usually  supported 
in  the  interstices  of  massive  grids  of  metallic  lead.     These  lead 
grids  serve  not  only  as  mechanical   supports 
for  the  active  material,  but  they  serve  also 
to  deliver  current  to  or  receive  current  from 
the  active  materials  which  constitute  the  real 
electrodes. 

Figure  io£  shows  a  commercial  form  of 
lead  storage  cell.  The  electrodes  consist  of 
fine  grids  of  metallic  lead  in  the  interstices 
of  which  the  active  material  is  placed.  The 
positive  electrode  (out  of  which  the  current 
comes  during  discharge)  consists  of  three 
grids  connected  together,  and  the  negative  electrode  consists  of 
four  grids  connected  together. 

Action  of  the  cell  while  discharging. —  When  the  lead  storage  cell  delivers  cur- 
rent, the  electrolyte  H2SO4  is  split  up  by  the  current  into  H2  and  SO4.  The 
hydrogen  is  liberated  at  the  cathode,  where  it  reduces  the  lead  peroxide  to  PbO,  and 
this  PbO  combines  with  a  portion  of  the  H2SO4  of  the  electrolyte  forming  PbSO4 
and  water.  The  SO4  which  is  liberated  at  the  anode  combines  with  the  spongy 
lead  and  forms  PbSO4.  During  this  process  the  active  material  expands,  because  the 
lead  sulphate  is  more  bulky  than  the  spongy  lead  and  the  lead  peroxide  ;  and  the 
electrolyte  grows  less  concentrated  (and  of  course  increases  in  resistance)  because  of 
the  absorption  of  SO4  by  the  active  material.  This  decrease  of  concentration  is 
especially  great  in  the  pores  of  the  active  material  when  the  cell  is  discharged 
rapidly. 

Action  of  the  cell  while  being  charged.  —  When  the  lead  storage  cell  is  regenerated 
by  forcing  a  reversed  current  through  it,  the  above-described  action  is  reversed.  The 
lead  sulphate  on  one  electrode  is  converted  back  to  lead  peroxide,  the  lead  sulphate 
on  the  other  electrode  is  reduced  to  spongy  metallic  lead,  the  electrolyte  grows  more 
dense  (especially  in  the  pores  of  the  active  material),  and  the  active  material 
contracts. 

The  following  tabular  arrangement  gives  a  clear  idea  of  the  action  of  the  lead  stor- 
age cell  while  discharging  and  while  being  charged  : 


Positive  grid. 
Negative  grid. 


DISCHARGING. 
PbO,   +H2S04 


Pb 


H2  =r2H20     +PbS04 

A.     Direction  of  current  through  the  cell 
|||        (negative  to  pos  live  grid). 

S04  =  PbSO4 


20          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

CHARGING. 


Positive  grid.*  PbSO4  +  2H2O  +  S04  =  2H2SO4  +  PbO2 

$    Direction  of  current  through  the  ceil 
y        (positive  to  negative  grid). 

Negative  grid.*         PbSO4      ,+  H,  =  H2SO4   +  Pb 

10.  Open-circuit  cells  and  closed-circuit  cells.  —  A  voltaic  cell 
in  which  the  local  action  is  very  slight  does  not  deteriorate 
appreciably  when  it  is  not  called  upon  to  deliver  current.  Such 
a  cell  may  be  left  standing  on  open  circuit  in  readiness  for  use  at 
any  moment  to  supply  current  for  any  purpose  such  as  to  ring  an 
electric  bell.  All  that  is  necessary  is  to  provide  a  device  for 
closing  the  circuit  when  it  is  desired  to  obtain  current  from  the 
cell,  and  then  the  circuit  should  be  opened  in  order  to  avoid 
deterioration  of  the  cell  by  the  continued  flow  of  current.  A 
voltaic  cell  which  is  adapted  to  this  kind  of  use  is  called  an  open- 
circuit  cell  and  perhaps  the  best  form  of  open-circuit  cell  is  the 
ordinary  dry  cell. 

When  an  ordinary  dry  cell  is  called  upon  to  give  a  steady  cur- 
rent the  electromotive  force  f  falls  off  rapidly  on  account  of  what 
is  called  polarization,  and  the  current  decreases  accordingly.  A 
voltaic  cell  which  is  capable  of  delivering  a  fairly  large  steady 
current  is  called  a  closed-circuit  cell.  The  gravity  cell  is  one  of 
the  best  types  of  closed-circuit  cell.  The  chromic  acid  cell  is  also 
frequently  used  for  delivering  current  more  or  less  steadily.  The 
Edison-LaLande  cell  is  a  fairly  good  open-circuit  cell  and  it  is 
satisfactory  also  for  closed-circuit  work. 

PROBLEMS. 

1.  The  anode  of  an  electrolytic  cell  consists  of  a  copper  rod 
3  centimeters  in  diameter,  and  the  cathode  consists  of  a  hollow 

*  It  is  the  usual  practice  among  electrical  engineers  to  call  that  terminal  of  an 
electric  generator  out  of  which  current  flows,  the  positive  terminal,  and  that  terminal 
into  which  current  flows,  the  negative  terminal.  In  conformity  with  this  usage,  that 
electrode  of  a  storage  cell  which  is  cathode  during  discharge  is  called  the  positive  grid 
and  the  other  the  negative  grid.  The  positive  grids  are  of  a  pale  salmon  color  and 
the  negative  grids  are  a  neutral  gray. 

f  See  Art.  22. 


THE   ELECTRIC   CURRENT.  21 

copper  cylinder  of  which  the  inside  diameter  is  1 2  centimeters. 
1 5  centimeters  of  length  of  anode  and  cathode  are  submerged  in 
the  electrolyte,  and  a  current  of  25  amperes  is  passed  through 
the  cell,  (a)  Find  the  current  density  at  the  cathode,  and  (b) 
find  the  current  density  at  the  anode.  Ans.  (a)  0.044  ampere 
per  square  centimeter;  (b)  0.177  ampere  per  square  centimeter. 
2.  An  electric  current  is  sent  through  an  ammeter  and  through 
a  silver  coulombmeter.  The  current  gives  a  steady  reading  of 
1. 068  amperes  on  the  ammeter,  and  the  amount  of  silver  depos- 
ited in  I  hour  and  20  minutes  is  found  by  weighing  to  be  5.635 
grams.  Find  the  error  of  the  ammeter  reading.  Ans.  0.018 
ampere  too  high. 

Note.  — The  silver  coulombmeter  is  usually  arranged  as  shown  in  Fig.  II.     The 
silver  nitrate  solution  is  contained  in  a  clean  platinum  bowl  which  serves  as  the  cathode 
on  the  interior  of  which  the  silver  is 
deposited,  and  the  anode  consists  of  a 
plate  of  pure  silver  surrounded  by  a 
covering  of  filter  paper  to  prevent  de- 
tached  particles  from    falling  to  the  \  :==^>/f>s^r^r  jpfatinum  bowl 
bottom  of  the  platinum  bowl. 


3.  Calculate    the   electro- 


chemical equivalents  of  the  sheet  of  metal 

c  11         •  /    \    /-  FiS-  !  !• 

following  :  (a)  Cuprous  cop- 
per ;  (&)  cupric  copper  ;  (c)  zinc  ;  (d)  hydrogen  ;  (e)  aluminum  ; 
and  (/)  ferric  iron.  The  valencies  of  the  respective  metals  may 
be  inferred  from  the  following  formulae  of  their  chlorides  :  (a) 
CuCl;  (£)CuCl2;  (V)ZnCl2;  (d)  HC1 ;  (e)  A1C13 ;  (/)  FeCl3 
Ans.  (a)  0.0006587  gram  per  ampere  per  second  ;  (b}  0.0003293 
gram  per  ampere  per  second  ;  (c)  0.0003 39  gram  per  ampere  per 
second;  (d}  0.00001046  gram  per  ampere  per  second;  (e\ 
0.0000936  gram  per  ampere  per  second ;  (/)  0.0001929  gram 
per  ampere  per  second. 

4.  A  current  which  gives  a  steady  reading  of  10  amperes  on 
an  ammeter  is  found  to  deposit  8.24  grams  of  copper  in  40  min- 
utes from  a  solution  of  CuSO4.  What  is  the  error  of  the  ammeter 
reading  ?  Ans.  0.42  ampere  too  low. 


22  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

5.  Find  the  time  required  for  10  amperes  of  current  to  gener- 
ate   2    cubic    feet  of    hydrogen  and    i    cubic    foot  of  oxygen, 
both  gases  being  measured  at  730  millimeter  pressure  and  at  a 
temperature  of  20°  C.  ;   22  millimeters  of  the  pressure  in  each 
case  being  due  to  the  water  vapor  which  is  present.     Ans.  n.8 

hours. 

, 

Note.  —  The  amount  of  hydrogen  or  oxygen  generated  in  one  second  by  one 
ampere  may  be  found  from  the  electrochemical  equivalent  of  silver  and  the  atomic 
weights  of  hydrogen,  oxygen  and  silver,  or  the  data  given  in  the  note  to  problem  6 
may  be  used. 

6.  A  current  which  produces  a  steady  reading  of  5  amperes  on 
an  ammeter  generates  186.6  cubic  centimeters  of  a  mixture  of 
oxygen  and  hydrogen  in  3  minutes,  the  mixed  gases  being  meas- 
ured at  a  net  pressure  of  710  millimeters  and  at  a  temperature  of 
25°    C.     Find   the  error  of  the  ammeter  reading.     Ans.   o.i 
ampere  too  low. 

Note.  —  By  net  pressure  in  this  problem  is  meant  the  pressure  due  to  the  gas  alone 
after  correction  has  been  made  for  the  part  of  the  pressure  which  is  due  to  the  water 
vapor  that  is  present. 

The  water  coulombmeter  is  frequently  used  for  quickly  standardizing  an  ammeter, 
and  it  is  convenient  to  note  that  one  ampere  in  one  second  generates  o.  174  cubic  centi- 
meter of  mixed  hydrogen  and  oxygen,  the  gases  being  measured  dry,  at  760  milli- 
meters pressure,  and  at  a  temperature  of  o°  C. 

7.  A  voltaic  cell  which  is  free  from  local  action  gives  a  current 
of  1.5  amperes  for  50  hours.     Calculate  the  number  of  grams  of 
zinc  consumed.     Ans.  91.5  grams. 

Note.  —  The  zinc  consumed  in  a  voltaic  cell  by  voltaic  action  is  equal  to  the  amount 
of  zinc  that  would  be  deposited  in  an  electrolytic  cell  by  the  current  which  the  cell 
delivers. 

8.  A  single  chromic  acid  cell  consumes   125  grams  of  zinc 
during  the  time  that  the  current  from  the  cell  is  depositing  25 
grams  of  copper  from   a  solution  of  cupric  sulphate  (CuSOJ. 
What  portion  of  the   zinc  is  consumed  by  local  action  ?     Ans. 
79.4  per  cent. 

9.  A  gravity  cell  is  used  to  give  a  steady  current  of  o.  I  ampere 
continuously,  night   and    day,  for  30  days.     During   this  time 


THE   ELECTRIC    CURRENT.  23 

1 668. 6  grams  of  copper  sulphate  crystals  are  used.  Find:  (a) 
The  amount  of  copper  sulphate  crystals  which  is  consumed  by 
voltaic  action,  and  (b)  the  amount  of  copper  sulphate  crystals 
which  is  consumed  by  local  action.  Ans.  (a)  504.6  grams  use- 
fully consumed  in  voltaic  action  and  (<£)  1,164  grams  wasted  in 
local  action. 

Note.  —  Copper  sulphate  crystals  contain  12  molecules  of  water  of  crystallization, 
that  is  to  say,  the  formula  for  copper  sulphate  crystals  is  CuSO4  -|-  !2HaO  so  that 
375-9  grams  of  copper  sulphate  crystals  contain  63.6  grams  of  copper. 

10.  An  ordinary  dry  cell  was  connected  to  a  circuit,  the  cur- 
rent at  the  start  was  5.00  amperes,  and  the  current  was  observed 
at  intervals  of  I  o  minutes,  giving  the  following  values  in  amperes 
in  order:  4.20,   3.92,  3.70,   3.55,   3.40,   3.28,  3.16,  3.02,  2.90, 
2.81,  2.72,  2.60,  2.54,  2.48,  2.43,  2.37,  2.30,  2.24,2.16.     Plota 
curve  of  which  the  abscissas  represent  elapsed  times  and  of  which 
the  ordinates  represent  the  decreasing  values  of  the  current  deliv- 
ered by  this  cell. 

11.  A  lead  storage  cell  delivers  10  amperes  for  8  hours.     Find  the  increase  of 
weight  of  each  electrode.     Ans.  The  positive  electrode  or  grid  gains  95.5  grams,  and 
the  negative  grid  gains  143.3  grams. 

12.  The  storage  cell  specified  in  problem  II  contains  4,000  cubic  centimeters  of 
dilute  sulphuric  acid  of  which  the  density  at  18°  C.  is  1.1700  grams  per  cubic  centi- 
meter when  the  cell   is  fully  charged.      Find  the  density  of  the  electrolyte  after  the 
cell  has  delivered  10  amperes  for  8  hours.      Ans.    1.1286  grams  per  cubic  centimeter. 

DATA  REQUIRED  IN  THE  ABOVE  PROBLEMS. 
ATOMIC  WEIGHTS. 

Silver 107.93  Sodium 23.05 

Aluminum 27.1  Oxygen 16.00 

Copper 63.6  Lead 206.91 

Iron 55-88  Sulphur 32.06 

Hydrogen l.oi  Zinc 65.40 

Density  of  dry  hydrogen  at  o°  C.  and  760  mm.  pressure,  0.0000896  gram  per  cubic 
centimeter. 

Density  of  dry  oxygen  at  o°  C.  and  760  mm.  pressure,  0.001429  gram  per  cubic 
centimeter. 


24  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

DENSITY  OF  DILUTE  SULPHURIC  ACID  IN  GRAMS  PER   CUBIC  CENTIMETER 

AT  1 8°  C. 

o  per  cent.  H2SO4  0.9986 

10  per  cent.  H2SO4  1.0673 

20  per  cent.  :^SSO4  1.1414 

30  per  cent.  H2SO4  1.221 

Per  cent,  of  H2SO4   in  this  table  means  the  number  of  grams  of  H2SO4   in   100 
grams  of  the  solution. 

The  electrochemical  equivalent  of  silver  is  0.001118  gram  per  ampere  per  second. 


CHAPTER   II. 
RESISTANCE  AND   ELECTROMOTIVE   FORCE. 

HEATING  EFFECT  OF  THE  ELECTRIC  CURRENT. 

11.  Electrical  resistance. — When  a  pump  forces  water  through 
a  circuit  of  pipe,  a  part  of  the  work  expended  in  driving  the  pump 
reappears  as  heat  in  the  various  parts  of  the  circuit  of  pipe  because 
of  the  resistance  which  the  pipe  offers  to  the  flow  of  water. 
Similarly,  when  an  electric  generator  produces  an  electric  cur- 
rent in  a  circuit,  a  part  of  the  work  expended  in  driving  the 
generator  reappears  as  heat  in  the  various  parts  of  the  circuit. 
The  current  seems  to  be  opposed  by  a  kind  of  resistance  *  more 
or  less  analogous  to  the  resistance  which  a  pipe  offers  to  the 
flow  of  water,  and  a  portion  of  an  electrical  circuit  is  said  to  have 
more  or  less  electrical  resistance  according  as  more  or  less  heat  is 
generated  in  it  by  a  given  current. 

12.  The  heating  effect  of  the  electric  current.     Joule's  law.  — 

The  amount  of  heat  which  is  generated  in  a  given  wire  is  propor- 
tional to  the  square  of  the  current  that  is  flowing  in  the  wire  and 
to  the  time  that  the  current  continues  to  flow,  that  is, 

H=RPt  (2) 

in  which  H  is  the  amount  of  heat  generated  in  a  wire  in  /  sec- 
onds by  a  current  of  /  amperes,  and  R  is  a  constant  for  a 
given  wire.  The  value  of  this  factor  R  is  used  as  a  numerical 
measure  of  the  electrical  resistance  of  the  wire. 

Practical  applications  of  the  heating  effect.  —  The  heating  effect 
of  the  electric  current  is  utilized  in  the  various  forms  of  electric 
lamps  in  which  a  filament  of  carbon  or  refractory  metal  is  heated 
to  brilliant  incandescence  by  the  electric  current.  The  heating 

*  An  exact  mechanical  analogue  of  electrical  resistance  is  given  in  Art.  62. 

25 


26  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

effect  of  the  electric  current  is  also  utilized  in  a  variety  of  electric 
furnaces.  * 

Definition  of  the  ohm.  —  If  H  in  equation  (2)  is  expressed  in 
joules, f  /  in  amperes,  and  t  in  seconds,  then  R  is  expressed 
in  terms  of  a  unit  which  is  called  the  ohm,  that  is,  a  wire  has  one 
ohm  of  resistance  when  one  joule  of  heat  is  generated  in  it  in 
one  second  by  one  ampere  of  current.  The  meaning  of  the 
factor  R  in  equation  (2)  may  be  made  clear  by  solving  this 
equation  for  R,  which  gives  R  =  Hj Izt.  According  to  this 
equation,  the  resistance  of  a  wire  in  ohms  is  equal  to  the  joules 
of  heat  generated  in  it  per  ampere  squared  per  second,  or  in 
other  words,  an  ohm  is  one  joule-per-ampere-squared-per-second. 
The  abohm  is  defined  in  Art.  52. 

The  international  standard  ohm.  —  The  resistance  of  a  wire  or 
other  portion  of  an  electrical  circuit  can  be  measured  with  great 
ease  in  terms  of  a  known  resistance,  whereas  a  fundamental 
measurement  of  resistance  requires  elaborate  arrangements,  and  it 
is  very  tedious  if  a  moderate  degree  of  accuracy  is  desired. 
Therefore,  for  practical  purposes,  the  ohm  has  been  legally 
defined  J  as  the  resistance  at  the  temperature  of  melting  ice  of  a 
column  of  pure  mercury  106.3  centimeters  long,  of  uniform  cross- 
sectional  area,  and  weighing  14.4521  grams. 

Measurement  of  resistance.  —  A  direct  method  for  measuring 
the  resistance  of  a  wire  is  to  send  a  known  current  /  through  the 
wire  for  a  known  length  of  time  /  and  to  determine  the  amount 
of  heat  generated  in  the  wire  by  means  of  a  water  calorimeter. 
This  direct  method  for  measuring  the  resistance  of  a  wire  in 

*  See  Calcium  Carbide  Manufacture  at  Niagara,  Electrochemical  Industry,  Vol. 
I,  page  22,  and  Carborundum  Manufacture  at  Niagara,  Electrochemical  Industry, 
Vol.  I,  page  50.  See  report  of  Canadian  Commission  on  Electrothermic  Processes 
for  the  Smelting  of  Iron  and  Steel,  by  Eugene  Haanel. 

f  Ordinarily  heat  is  expressed  in  terms  of  the  calorie  but  it  is  desirable  in  the 
present  instance  to  express  h£at  in  joules,  one  joule  of  heat  being  an  amount  of  heat 
which  is  equivalent  to  one  joule  of  work.  One  calorie  is  equal  to  4.2  joules.  One 
joule  of  work  is  the  amount  of  work  done  in  one  second  by  an  agent  which  does  work 
at  the  rate  of  one  watt.  One  watt  is  equal  to  1/746  of  a  horse-power. 

J  In  accordance  with  the  recommendations  of  the  International  Electrical  Congress 
which  met  at  Chicago  in  1893. 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  27 

ohms  is  never  used  because  it  is  tedious  and  inaccurate.  Practi- 
cal methods  for  measuring  resistance  are  described  in  Chapter  X. 

13.  Power  required  to  maintain  a  current  in  a  circuit,  expressed 
in  terms  of  resistance  and  current.  —  When  all  of  the  energy 
which  is  delivered  to  an  electrical  circuit  by  a  generator  reappears 
in  the  circuit  as  heat,  then  the  rate  at  which  work  is  delivered  to 
the  circuit  by  the  generator  is  equal  to  the  rate  at  which  energy 
reappears  in    the   circuit  as   heat.      Equation  (2)  expresses  the 
amount  of  heat  in  joules  which  appears  in  a  circuit  of  wire  in    / 
seconds  ;  dividing  this  amount  of  heat  by  the  time    t,    gives  the 
rate  at  which  heat  appears  in  the  circuit  in  joules  per  second 
(watts),  and  this  is  equal  to   RP.     Therefore  the  power   Pt    in 
watts,  required  to  maintain  a  current  of  /  amperes  in  a  circuit 
of  which  the  resistance  is    R   ohms,  is 

P=RP  (3) 

14.  Dependence  of  resistance  upon  length  and  size  of  a  wire.  — 

The  resistance  R  of  a  wire  of  given  material  is  directly  propor- 
tional to  the  length  /  of  the  wire  and  inversely  proportional  to 
the  sectional  area  s  of  the  wire ;  that  is, 

R-*1-  (4) 

in  which  k  is  a  constant  for  a  given  material ;  it  is  called  the 
resistivity  *  of  the  material.  The  exact  meaning  of  the  factor  k 
may  be  made  apparent  by  considering  a  wire  of  unit  length 
(/=  i)  and  unit  sectional  area  (5=1).  In  this  case  k  is 
numerically  equal  to  R,  that  is  to  say,  the  resistivity  of  a 
material  is  numerically  equal  to  the  resistance  of  a  wire  of  that 
material  of  unit  length  and  unit  sectional  area.  Electrical  engi- 
neers nearly  always  express  lengths  of  wires  in  feet  and  sectional 
areas  in  circular  mils.f  If  equation  (4)  is  to  be  used  to  calculate 

*  Sometimes  called  specific  resistance.  The  reciprocal  of  the  resistivity  of  a  sub- 
stance is  called  its  conductivity. 

f  One  mil  is  a  thousandth  of  an  inch.  One  circular  mil  is  the  area  of  a  circle  of 
which  the  diameter  is  one  mil.  The  area  of  any  circle  in  circular  mils  is  equal  to  the 
square  of  the  diameter  of  the  circle  in  mils. 


28 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


the  resistance  of  a  wire  in  ohms  when  the  length  of  the  wire  is 
expressed  in  feet  and  the  sectional  area  in  circular  mils,  then  the 
value  of  k  must  be  the  resistance  of  a  wire  of  the  given  material 
one  foot  long  and  one  circular  mil  in  sectional  area  ;  for  example, 
the  resistance  of  a  copper  wire  one  foot  long  and  one  circular  mil 
in  sectional  area  is  about  10.4  ohms  at  20°  C. 


TABLE. — RESISTIVITIES  AND  TEMPERATURE  COEFFICIENTS. 


a 

b 

C 

Aluminum  wire  (annealed)  at  20°  C. 

27  4    VlO~ 

16  c 

_I_O  OO3Q 

Copper  wire  (annealed)  at  20°  C 

17  24VIO" 

*^o 

IO  4. 

-J-O  0040 

Iron  wire  (pure  annealed)  at  20°  C 

QC           VlO~ 

eg  O 

-4-O  OO41 

Steel  telegraph  wire  at  20°  C 

ICQ      Vio~ 

Qlt 

•4-O  OO431" 

Steel  rails  at  20°  C    

1  20      Vio~ 

72t 

-4-O  OO^^t 

Mercury  at  o°  C  ... 

043.4  Vicr 

-j-o  00088 

Platinum  wire  at  o°  C  

8q.8  Vicr 

t!4  O 

-j-O  OO1Z4. 

German-silver  wire  at  20°  C  

212      Vicr 

I27t 

-L  o  OOO2  S  t 

Manganin  wire  (Cu  84,  Ni  12,  Mn  4)  at  20°  C... 
"la  la"  metal  wire,  hard   (copper-nickel  alloy) 
at  20°  C 

475      Xio- 
coo      V  io~7 

286 

•loot 

* 

o  ooooi  i 

"Climax"    or    "Superior"    metal    (nickel-steel 
alloy)  at  20°  C  

800      Vicr7 

4.8ot 

-J-  o  0006  7  i 

Arc-lamp  carbon  at  ordinary  room  temperature  ... 
Sulphuric  acid,  5  per  cent,  solution  at  1  8°  C  
Ordinary  glass  at  o°  C.  (density  2.54)  

0.005 
4.8  ohms 
IO15  ohmsj 

—  0.0003! 

—  O.OI2Of 

Ordinary  glass  at  60°  C  

IO12  ohmsj 

Ordinary  glass  at  200°  C 

lo8  ohmsj 

a  =  resistance  in  ohms  of  a  bar  I  centimeter  long  and  I  square  centimeter  sec- 
tional area. 

6  =  resistance  in  ohms  of  a  wire  I  foot  long  and  o.ooi  inch  in  diameter. 

c  =  temperature  coefficient  of  resistance  per  degree  centigrade  (mean  value  be- 
tween o°  C.  and  100°  C. ). 

*  See  temperature-resistance  curve,  Fig.   15. 

t  Between  18°  C.  and  19°  C. 

\  These  values  differ  greatly  with  different  samples. 

15.  Resistivities  of  alloys,  —  The  ordinates  of  the  three  curves 
in  Fig.  1 2  represent  the  resistivities  at  a  given  temperature  of 
alloys  of  zinc  and  tin,  of  silver  and  gold,  and  of  silver  and  plati- 
num, respectively,  and  the  abscissas  represent  the  percentages  of 
the  constituent  metals.  The  zinc-tin  line,  marked  Zn  +  Sn,  is 
sensibly  straight ;  that  is,  the  change  of  resistance  from  pure  zinc 
to  pure  tin  is  proportional  to  the  percentage  of  tin  in  the  alloy. 


RESISTANCE   AND    ELECTROMOTIVE   FORCE. 


29 


The  silver-platinum  line  marked  Ag  +  Pt,  and  the  silver-gold 
line,  marked  Ag  -f  Au,  are  not  straight.  In  particular,  it  is  to 
be  noticed  that  a  very  small  percentage  of  platinum  added  to  pure 
silver  increases  the  resistance  of  the  metal  very  greatly  indeed. 


10 


ehl- 


4 

Zn=3.68 


Ag-1.00 


PERCENTAGE  -(-COMPOSITION 


\ 


\ 


Sn=8.35 


Au-1.28 


20  40  60 

Fig.  12. 

In  respect  to  electrical  resistance,  the  alloys  of  tin,  lead,  cad- 
mium and  zinc  are  similar  to  the  alloys  of  zinc  and  tin,  that  is  to 
say,  the  resistivity  varies  in  proportion  to  the  percentage  of  one 
of  the  metals  in  the  alloy.  Alloys  of  most  other  metals  are  more 
or  less  similar  to  the  alloys  of  silver  and  gold  and  of  silver  and 
platinum,  and,  in  general,  the  addition  of  a  very  small  percentage 
of  one  metal  to  another  increases  the  resistivity  greatly.  The 
exact  opposite  to  this  is  true  of  many  non-metallic  substances,  a 


30  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

pure  substance  has  a  very  high  resistance  and  the  admixture  of  a 
very  small  quantity  of  another  substance  reduces  its  resistance 
very  greatly  indeed.  Thus  a  beaker  of  freshly  distilled  water  (free 
from  air)  in  which  are  placed  twcwclean  platinum  electrodes  has  a 
resistance  of,  say,  25,000  ohms  and  the  addition  of  one  one-thou- 
sandth of  one  per  cent,  of  sulphuric  acid  reduces  the  resistance  to 
a  few  hundreds  of  ohms. 

16.  The  rheostat.  —  An  arrangement  for  inserting  more  or  less 
resistance  into  an  electrical  circuit  at  will  is  called  a  rheostat. 
Figure  1 3  shows  the  usual  arrangement  of  a  rheostat.  A  number 
of  resistances  rrrr  are  connected  to  terminal  blocks  of  metal  bbbbb 
and  a  contact  finger  /  of  metal,  broad  enough  to  bridge  over  the 
space  between  the  adjacent  blocks  bbt  is  arranged  so  that  it  can 


Fig.  13. 

be  moved  sidewise,  thus  connecting  any  number  of  resistances  r 
in  circuit  between  the  terminals  A  and  B  of  the  rheostat.  The 
resistances  rrrr,  Fig.  13,  are  usually  made  of  metal  of  high 
specific  resistance  so  that  the  wire  may  be  of  moderate  length 
and  yet  large  enough  to  be  mechanically  strong  and  to  have  suf- 
ficient area  to  radiate  the  heat  which  is  generated  in  it  by  the 
current.  One  of  the  most  satisfactory  of  these  high  resistance 
metals  is  a  nickel-steel  alloy  which  is  known  in  commerce  under 
the  name  of  "  Climax  "  metal  or  "  Superior  "  metal. 

The  so-called  water  rheostat  which  is  frequently  used  consists 
of  two  electrodes  dipping  into  a  vessel,  or  tank,  containing  a  weak 
solution  of  common  salt.  The  current  enters  at  one  electrode, 
flows  through  the  salt  solution,  and  leaves  it  at  the  other  electrode, 
and  the  resistance  can  be  adjusted  by  varying  the  amount  of  salt 
in  solution  or  by  moving  the  electrodes. 


RESISTANCE   AND    ELECTROMOTIVE   FORCE. 


17.  Variation  of  resistance  with  temperature.  —  The  electrical 
resistance  of  a  wire,  or  of  a  liquid  column  which  forms  a  portion 
of  an  electrical  circuit,  varies  with  temperature.  Consider,  for 
example,  (a)  an  iron  wire,  (<£)  a  copper  wire,  (c)  a  platinum  wire, 
(d)  a  german-silver  wire,  (e)  a  carbon  rod,  and  (/)  a  column  of 
dilute  sulphuric  acid,  each  of  which  has  a  resistance  of  100  ohms 


ohms 


i  Be 
j6o 
140 

120 

100 

80 
60 


o          20         40         60        .8a        too        120        140        160       180        200 
degrees  centigrade 

Fig.  14. 

at  0°  C.  The  values  of  the  resistance  of  (a),  (b\  (c\  (d\  (e)  and 
(/)  at  other  temperatures  are  shown  by  the  ordinates  of  the 
curves  in  Fig.  14.  It  is  evident  from  Fig.  14  that  iron  and  cop- 
per increase  very  greatly  in  resistance  with  rise  of  temperature, 
and  that  german  silver  increases  slightly,  whereas  the  carbon  and 
sulphuric  acid  decrease  in  resistance  with  rise  of  temperature. 
All  pure  metals  increase  in  resistance  with  rise  of  temperature  in 
approximately  the  same  ratio,  alloys  usually  increase  in  resistance 
with  rise  of  temperature  but  to  a  much  smaller  extent  than  pure 


32          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

metals,  and  all  acids  and  salt  solutions  decrease  in  resistance  with 
rise  of  temperature. 

A  rod  of  a  substance  like  glass  or  porcelain  has,  at  ordinary 
room  temperature,  a  resistance  which  is  expressed  in  millions  of 
millions  of  ohms,  but  the  resistance  decreases  rapidly  with  rise  of 
temperature.  Both  glass  and  porcelain  become  fairly  good  con- 
ductors at  a  low  red  heat.  This  is  strikingly  shown  by  the  fol- 
lowing experiment :  Fine  copper  wire  is  wound  around  the  ends 
of  a  thin-walled  glass  tube  about  20  centimeters  long,  and  these 
copper  wires  are  connected  through  a  fairly  high  metallic  resist- 
ance to  the  terminals  of  a  i,ooo-volt  transformer.  The  side  of 
the  tube  is  then  heated  with  a  blast  lamp.  At  a  low  red  heat  a 
sufficient  amount  of  current  begins  to  flow  to  develop  a  very  con- 
siderable amount  of  heat,  and  the  glass  tube  becomes  still  hotter, 
which  permits  still  more  current  to  flow,  which  makes  the  glass 
tube  still  hotter,  and  so  on,  until  the  glass  tube  melts  down  be- 
cause of  the  heat  which  is  generated  in  it  by  the  flow  of  current. 

Alloys  which  change  but  little  in  resistance  with  change  of 
temperature  are  especially  suitable  for  resistance  standards  and 
resistance  boxes.  *  Wires  of  manganin  f  are  now  almost  uni- 
versally employed  for  this  purpose.  Figure  1 5  J  shows  the 
change  of  resistance  of  a  manganin  wire  with  temperature.  A 
manganin  wire  which  has  a  resistance  of  I  oo  ohms  at  1 5  °  C.  has 
a  resistance  of  100.01  ohms  at  20°  C.  ;  a  german-silver  wire 
which  has  a  resistance  of  100  ohms  at  15°  C.,  has  about  100.2 
ohms  resistance  at  20°  C.;  and  a  copper  wire  which  has  resistance 
of  100  ohms  at  15°  C.,  has  about  102  ohms  resistance  at  20°  C.; 
that  is,  for  the  specified  rise  of  temperature  the  change  of  resist- 
ance of  the  manganin  wire  is  only  o.oi  per  cent.,  the  change  of 
resistance  of  the  german-silver  wire  is  0.2  per  cent.,  and  the 
change  of  resistance  of  the  copper  wire  is  2  per  cent. 

*  See  Chapter  X. 

t  Manganin  is  an  alloy  of  84  parts  by  weight  of  copper,  12  parts  by  weight  of 
nickel,  and  4  parts  by  weight  of  manganese. 

|  From  the  results  of  Dr.  Lindeck.  See  the  Proceedings  of  the  International 
Electrical  Congress,  Chicago,  1893,  page  165. 


RESISTANCE   AND    ELECTROMOTIVE    FORCE. 


33 


Temperature  coefficient  of  resistance.  —  The  curves  a,  b  and  c 
in  Fig.  1 4  are  approximately  straight  lines  ;  the  same  is  true  of  the 
temperature-resistance  curves  of  all  pure  metals  and  of  many 
alloys.  Therefore,  the  increase  of  resistance  of  a  wire  from  a 
standard  temperature,  say,  o°  C.,  to  any  other  temperature  /°  C. 


ohms 
100,03 

100.02 
IOO.OI 

100.00 

•b* 

*~  - 

•\ 

/ 

/ 

N, 

K 

/ 

\ 

V 

/ 

1 

/ 

10           20           30           40           50           60           70           Be 
degrees  centigrade 

Fig.  15. 

is  approximately  proportional  to  /,  and  in  every  case  the  increase 
of  resistance  is  exactly  proportional  *  to  the  resistance  of  the  wire 
at  the  standard  temperature.  Therefore  the  increase  of  resist- 
ance from  oc  C.  to  t°  C.  may  be  expressed  as  @RQt,  where 
RQ  is  the  resistance  of  the  wire  at  o°  C.  and  /3  is  a  factor 
which  is  approximately  constant  for  a  given  metal.  The  resistance 
of  the  wire  at  t°  C.  is  equal  to  RQ  -f-  @RQt,  so  that,  writing 
Rt  for  the  resistance  of  the  wire  at  t°  C.,  we  have 


+Bt) 


(5) 


*  This  is  analogous  to  the  fact  that  the  increase  of  length  of  a  metal  bar  due  to  a 
given  rise  of  temperature  is  exactly  proportional  to  the  initial  length  of  the  bar.  Con- 
sider for  example,  a  bar  10  feet  long  'at  o°  C.  When  the  temperature  is  increased, 
each  foot  of  the  bar  increases  its  length  by  a  certain  fractional  part  of  a  foot,  and  the 
entire  bar  increases  its  length  by  the  same  fractional  part  of  its  total  initial  length. 

4 


34          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  factor  ft  is  called  the  temperature  coefficient  of  resistance 
of  the  given  metal.  It  is  equal  to  the  increase  of  resistance  of 
the  metal  for  one  degree  rise  in  temperature  expressed  as  a  frac- 
tional part  of  the  resistance  of  the  metal  at  o°  C.  Its  value  for 
pure  metals  is  approximately  0.0037  Per  degree  centigrade.  For 
pure  commercial  copper  wire  its  value  is  about  0.004  per  degree 
centigrade. 

It  is  to  be  remembered  that  equation  (5)  is  based  on  the  as- 
sumption that  the  temperature-resistance  curve  is  a  straight  line. 
If  the  actual  resistances  of  any  wire  or  substance  at  o°  C.  and 
at  /°  C.  are  substituted  in  equation  (5)  for  RQ  and  Rt,  respec- 
tively, the  value  of  ft  may  be  calculated.  The  value  of  ft  so 
calculated  is  called  the  mean  temperature  coefficient  of  resistance 
of  the  given  substance  for  the  given  range  of  temperature. 

The  value  of  the  temperature  coefficient  of  a  substance  depends 
upon  the  choice  of  the  standard  temperature  in  a  way  that  may 
be  most  easily  explained  by  considering  the  thermal  expansion 
of  a  gas.  A  gas  at  constant  pressure  undergoes  a  certain  definite 
increment  of  volume  for  one  degree  rise  of  temperature.  Thus, 
a  gas  at  constant  pressure  undergoes  the  same  increment  of  volume 
when  heated  from  10°  C.  to  11°  C.,  or  when  heated  from  50°  C. 
to  51°  C.,  or  when  heated  from  200°  C.  to  201°  C.  This  incre- 
ment of  volume  per  degree  rise  of  temperature  is  equal  to  ^-%  of 
the  volume  of  the  gas  at  o°  C.,  to  ^^  of  the  volume  of  the  gas 
at  i°  C,  to  3^3-  of  the  volume  of  the  gas  at  100°  C.,  and  so  on, 
and  this  fraction  is  the  temperature  coefficient  of  expansion  of 
the  gas.  In  order  to  avoid  ambiguity,  the  increment  of  volume 
of  a  gas  for  I  °  rise  of  temperature  is  always  expressed  as  a  frac- 
tional part  of  the  volume  of  the  gas  at  o°  C.,  and  the  coefficient 
of  expansion  of  a  gas  at  constant  pressure  is  therefore  equal  to 
^.^  (equals  0.00366).  Similarly,  the  temperature  coefficient  ot 
resistance  of  a  metal  should  always  refer  to  a  definite  standard 
temperature,  say,  o°  C.  It  is  interesting  to  note  that  the  temper- 
ature coefficient  of  resistance  of  most  pure  metals  is  very  nearly 
the  same  in  value  as  the  temperature  coefficient  of  expansion  of  a 


RESISTANCE   AND   ELECTROMOTIVE   FORCE.  35 

gas  at  constant  pressure.  That  is  to  say,  the  resistance  of  a  wire 
made  of  pure  metal  is  approximately  proportional  to  the  absolute 
temperature. 

ELECTROMOTIVE  FORCE. 

18.  Power  delivered  by  an  electric  generator  Definition  of 
electromotive  force.  —  From  Faraday's  laws  of  electrolysis  it  is 
evident  that  the  amount  of  zinc  consumed  per  second  in  a  voltaic 
cell  by  voltaic  action  is  proportional  to  the  strength  of  the  cur- 
rent. Therefore  the  available  *  energy  developed  per  second  by 
the  chemical  action  in  the  cell  is  proportional  to  the  strength  of 
the  current,  or  in  other  words,  the  electrical  energy  developed 
per  second  by  a  given  type  of  voltaic  cell  in  the  maintenance  of  a 
current  is  equal  to  a  constant  multiplied  by  the  current.  That  is, 

P=EI  (6) 

in  which  P  is  the  electrical  energy  developed  per  second  by  a 
voltaic  cell,  /  is  the  current  produced  by  the  cell,  and  E  is  a 
constant  for  the  given  type  of  cell.  This  constant  E  is  called 
the  electromotive  force  of  the  cell.  This  definition  of  electromotive 
force  applies  to  any  form  of  electric  generator.  Imagine  a 
dynamo  driven  at  constant  speed,  and  having  a  field  magnet  of 
which  the  strength  is  invariable.  Ignoring  friction,  the  only 
opposition  to  the  motion  of  the  dynamo  is  that  which  is  due  to 
the  current  flowing  through  the  armature  wires.  Therefore  to 
double  the  current  output  of  such  a  dynamo  would  double  the 
force  required  to  drive  it,f  and  therefore  double  the  rate  at  which 
work  would  be  expended  in  driving  it,  its  speed  being  constant ; 
but  the  work  which  would  be  expended  in  driving  such  a  dynamo 
would  all  go  to  maintain  the  current,  so  that  the  rate  at  which 

*  Available,  that  is,  for  the  production  of  current.  In  some  voltaic  cells  the  whole 
of  the  energy  developed  by  the  voltaic  action  goes  to  maintain  the  current ;  but,  in 
general,  a  definite  fractional  part  only  of  this  energy  is  available  for  the  production  of 
an  electric  current.  See  Physical  Chemistry,  H.  C.  Jones,  pages  376-405.  See  also 
papers  by  H.  S.  Carhart  "  On  the  Thermodynamics  of  the  Voltaic  Cell,"  Physical 
Review,  Vol.  XI,  p.  I,  Vol.  XVI,  p.  248,  and  Vol.  XXVI,  p.  209,  March,  1908. 

f  See  Art.  52>  on  the  side  push  of  a  magnetic  field  on  an  electric  wire. 


36          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

work  is  expended  in  maintaining  the  current  is  proportional  to  the 
current,  according  to  equation  (6). 

Hydraulic  analogue  of  electromotive  force.  —  An  electric  gen- 
erator, such  as  a  voltaic  cell  or  £  dynamo,  is  analogous  to  a  cen- 
trifugal pump,  or  fan  blower,  which  develops  a  definite  difference 
of  pressure  between  its  inlet  and  outlet.  Imagine  a  fan  blower 
connected  to  a  circuit  of  pipe  which  goes  out  from  the  outlet  and 
returns  to  the  inlet.  The  volume  of  air  per  second  forced  through 
this  pipe  may  be  called  the  strength  of  the  air  current,  and  the 
rate  at  which  the  fan  delivers  energy  in  the  maintenance  of  this 
air  current  is  equal  to  the  product  of  the  strength  of  the  air  cur- 
rent and  the  pressure  difference  between  inlet  and  outlet  of  the 
fan.  Let  /  be  the  strength  of  the  air  current  (volume  of  air 
flowing  per  second)  and  let  E  be  the  pressure  difference  between 
inlet  and  outlet.  The  power  developed  by  the  fan  in  maintaining 

the  flow  of  air  is 

P=EI 

This  equation  is  identical  to  equation  (6),  and  the  pressure  differ- 
ence between  inlet  and  outlet  of  the  fan  blower  is  exactly  analogous 
to  what  is  called  the  electromotive  force  of  an  electric  generator. 

Note.  —  The  power  delivered  by  a  fan  to  a  circuit  of  pipe  is  not  strictly  propor- 
tional to  the  volume  of  air  delivered  per  second  because  an  increased  flow  of  air  usually 
causes  a  slight  decrease  in  the  speed  of  the  fan.  Similarly,  the  power  delivered  to  a 
circuit  of  wire  by  a  voltaic  cell  or  dynamo  is  not  strictly  proportional  to  the  strength 
of  the  current  because  an  increase  of  current  usually  causes  a  decrease  in  the  electro- 
motive force  of  the  cell  or  generator.  This  decrease  of  electromotive  force  of  a  voltaic 
cell  is  called  polarization  and  it  is  discussed  in  Art.  22.  The  decrease  of  electromo- 
tive force  of  a  dynamo  due  to  increase  of  current  output  is  generally  due  to  a  slight 
decrease  of  speed  or  to  a  weakening  of  the  field  magnet,  or  to  both. 

Definition  of  the  volt.  —  When  P  in  equation  (6)  is  expressed 
in  watts  (joules  of  work  per  second)  and  /  in  amperes,  then  E 
is  expressed  in  terms  of  a  unit  which  is  called  the  volt.  That  is 
to  say,  the  electromotive  force  of  an  electric  generator  in  volts  is 
equal  to  the  power  in  watts  delivered  by  the  generator  divided 
by  the  current  in  amperes,  or  in  other  words,  the  power  deliv- 
ered by  a  generator  in  watts  is  equal  to  the  current  delivered  by 
the  generator  in  amperes  multiplied  by  the  electromotive  force  of 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  37 

the  generator  in  volts.     The  abvolt  or  c.g.s.  unit  of  electromotive 
force  is  defined  in  Art.  52. 

Unsatisfactory  character  of  the  fundamental  definition  of  electro- 
motive force.  —  The  definition  of  any  physical  quantity  consists,  in 
every  case,  of  a  concise  statement  of  the  fundamental  method  of 
measuring  that  quantity,  and  when  this  fundamental  method  of 
measuring  a  quantity  involves  operations  which  are  not  feasible 
under  ordinary  conditions  of  practical  work,  the  definition  seems 
more  or  less  unsatisfactory.  Thus,  the  above  definition  of  elec- 
tromotive force  as  units-of-work-per-second-per-ampere  (  P\I) 
assumes  that  the  rate  of  doing  work  in  a  pushing  current  through 
a  circuit  is  to  be  measured  directly  in  mechanical  units,  and  no 
method  is  specified  for  doing  this.  The  simplest  definition  of 
electromotive  force  is  based  on  Ohm's  Law  as  explained  in  the 
following  article. 

19.  Ohm's  Law.  —  The  current  produced  by  a  voltaic  cell,  or, 
in  general,  by  any  electric  generator,  is  inversely  proportional  to 
the  resistance  of  the  circuit.*  This  relation  was  discovered  by 
G.  S.  Ohm  in  1827  and  it  is  called  Ohm's  Law.  A  complete 
statement  of  Ohm's  Law  together  with  a  clear  specification  of  the 
conditions  under  which  the  law  applies  may  be  derived  as  fol- 
lows :  The  power  output  of  an  electric  generator  is  equal  to  El 
according  to  equation  (6).  If  the  whole  of  this  power  is  used  to 
heat  the  circuit  in  accordance  with  Joule's  Law,  then  we  must  have 


according  to  equation  (3).     Therefore  we  have 

E=RI  (70) 


or 


*This  statement  and  the  statement  given  in  the  previous  article  to  the  effect  that 
the  power  output  of  a  generator  is  proportional  to  the  current,  are  not  exactly  true, 
because  of  the  fact  that  the  electromotive  force  of  a  generator  usually  falls  off  in  value, 
to  some  extent,  when  the  generator  is  called  upon  to  give  an  increased  current. 


38  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Definition  of  the  volt  on  the  basis  of  Ohm's  Law.  —  According 
to  equation  (/^),  the  electromotive  force  required  to  force  a  cur- 
rent through  a  circuit  is  equal  to  the  product  of  the  resistance 
of  the  circuit  and  the  current.  When  the  resistance  is  expressed 
in  ohms  and  the  current  in  amperes,  this  equation  gives  the  value 
of  the  electromotive  force  in  volts.  That  is  to  say,  a  voltaic 
cell,  or  any  electric  generator  (assumed,  for  the  sake  of  simplicity 
of  statement  to  have  no  internal  resistance  and  to  be  unaffected 
by  those  secondary  influences  which  cause  a  decrease  of  electro- 
motive force  with  delivery  of  current),  has  an  electromotive  force 
of  one  volt  if  it  produces  one  ampere  of  current  in  a  circuit  of 
which  the  resistance  is  one  ohm. 

20.  Application  of  equations  (2),  (6)  and  (7)  to  a  portion  of  an 
electrical  circuit.  —  Equation  (2)  expresses  the  heat  which  is 
generated  in  a  portion  of  the  electrical  circuit,  R  being  the 
resistance  of  that  portion.  Equation  (6)  expresses  the  power 
which  is  delivered  to  a  portion  of  an  electrical  circuit,  E  being 
the  electromotive  force  across  the  terminals  of  that  portion. 
Equation  (7)  expresses  the  relationship  between  the  current  in  an 
electrical  circuit,  the  electromotive  force  across  any  given  portion 
of  the  circuit,  and  the  resistance  of  that  portion. 

The  current  produced  by  a  voltaic  cell  not  only  flows  through 
the  wire  which  is  connected  to  the  terminals  of  the  cell,  but  it 
flows  also  through  the  electrolyte  in  the  cell.  Let  EJ  repre- 
sent the  total  rate  at  which  work  is  supplied  by  the  voltaic  cell 
in  the  maintenance  of  the  current,  let  Rx  be  the  resistance  of  the 
external  circuit  of  wire,  and  let  Ra  be  the  resistance  of  the 
electrolyte  and  electrodes  in  the  cell.  Then  the  rate  at  which 
heat  is  generated  in  the  entire  circuit  is  (Ra  +  Rx)fz,  and  this  is 
equal  to  EJ,  so  that 


whence 

RJ-E.-RJ  (3) 

but   RJ  is  the  electromotive  force  which  is  required  to  force  the 
current    /  through  the  external  resistance    Rx  ;  that  is,    RJ  is 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  39 

the  actual  electromotive  force  between  the  terminals  of  the  cell  while 
it  is  delivering  current.  Therefore,  we  have 

EI  =  E,-RJ  (9) 

in  which  Ex  is  the  electromotive  force  across  the  terminals  of 
the  cell  while  it  is  delivering  current,  and,  inasmuch  as  Ex  —  RJt 
and  EJ  —  RXI2,  we  may  write  : 


and 


in  which  Px  is  the  power  delivered  by  the  cell  to  the  external 
circuit.  In  these  equations  Et  is  the  total  electromotive  force 
of  the  voltaic  cell  (or  generator),  RJ  is  the  portion  of  this  total 
electromotive  force  which  is  used  to  overcome  the  resistance  of 
the  cell  (or  generator),  Ex(  =  Et  —  RJ)  is  the  electromotive 
force  between  the  terminals  of  the  cell  (or  generator),  and  Px  is 
the  power  delivered  to  the  external  circuit  which  does  not  include 
the  power  developed  in  heating  the  cell  (or  generator). 

Equations    (6)    and    (7)    are  j  _  ^ 

nearly  always  used  in  practice  in         ( 
their  application  to  a  portion  of     ^IP^ 
a  circuit.     Thus,  Fig.  16  shows       ~^  - 
a  battery  B  supplying  current  to 

a  lamp  L,  the  electromotive  force  between  the  terminals  of  the 
lamp  is  E,  the  current  flowing  in  the  circuit  is  7,  the  power  de- 
livered to  the  lamp  is  El,  and  the  current  is  equal  to  the  elec- 
tromotive force  between  the  terminals  of  the  lamp  divided  by 
the  resistance  of  the  lamp,  according  to  equation  (7). 

Voltage  drop  in  a  generator.  —  The  electromotive  force  Ral 
required  to  overcome  the  resistance  of  the  generator  (or  voltaic 
cell)  in  the  above  discussion  is  subtracted  from  the  total  electro- 
motive force  of  the  generator  to  give  the  electromotive  force  be- 
tween the  generator  terminals,  as  indicated  in  equation  (8).  This 
electromotive  force  R  I  which  is  used  to  overcome  the  resistance 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


of  a  generator  is  called  the  electromotive  force  drop  or  voltage  drop 
in  the  generator.  It  is  analogous  to  the  decrease  of  pressure- 
difference  between  the  terminals  of  a  fan  blower  due  to  the  resist- 
ance which  is  encountered  by  the  stream  of  air  in  passing  through 
the  fan  chamber. 

Voltage  drop  in  a  transmission  line.  —  A  current  of  /  amperes 
is  delivered  to  a  distant  lamp  or  motor  over -a  pair  of  wires  the 
combined  resistance  of  which  is  R  ohms.  Let  J5Q  be  the  elec- 
tromotive force  across  the  terminals  of  the  generator,  and  let  E^ 
be  the  electromotive  force  across  the  terminals  of  the  distant  lamp. 


0 


O 


reference  axis 


n. 


large  return  pipe 


Fig.  17a. 

The  difference  between  the  voltage  across  the  terminals  of  the 
generator  and  the  voltage  across  the  terminals  of  the  lamp,  namely, 
EQ  —  El  is  equal  to  the  electromotive  force  which  is  used  to  over- 
come the  resistance  of  both  wires,  namely,  RI  volts.  This  loss 
of  electromotive  force  over  a  transmission  line  is  called  the  volt- 
age drop  over  the  line. 

Example.  —  The  electromotive  force  across  the  terminals  of  a 
generator  is  115  volts.  The  generator  supplies  100  amperes  of 
current  to  a  motor  at  a  distance  of  1,000  feet,  and  the  wire  (2,000 
feet)  used  for  the  transmission  has  a  total  resistance  of  0.05  ohm. 
The  voltage  drop  over  the  line  is  100  amperes  x  0.05  ohm,  or  5 
volts,  and  therefore  the  voltage  across  the  terminals  of  the  motor 
is  1 1 5  volts  —  5  volts  =  1 10  volts. 

Hydraulic   analogue   of  voltage  drop.     Definition  of  potential 


RESISTANCE   AND    ELECTROMOTIVE    FORCE.  41 

difference.  —  Figure  I  *ja  represents  a  pump  P  forcing  water 
through  a  small  pipe  and  through  a  distant  water  motor  M,  the 
water  being  returned  to  the  pump  through  a  very  large  and  ap- 
proximately frictionless  pipe.  The  motor  may  be  most  conven- 
iently thought  of  as  an  ordinary  pump  with  a  piston,  but  driven  as 
a  motor  by  the  water  which  is  forced  through  it  by  P.  Choosing 
the  pressure  in  the  large  pipe  as  zero  or  reference  pressure,  the 
pressure  at  any  other  point  in  the  system  is  to  be  specified  by 
giving  its  value  above  or  below  the  pressure  in  the  large  pipe. 
The  pump  draws  water  through  the  supply  pipe  s,  and  the  pres- 
sure in  this  small  pipe  falls  below  the  zero  line  or  axis  00.  At 
the  pump  there  is  a  sudden  rise  of  pressure  which  is  represented 
by  the  ordinate  A,  and  the  friction  of  the  long  pipe  causes  a 
steady  drop  of  pressure  until  the  motor  M  is  reached.  There  is 
a  sudden  drop  of  pressure  at  the  motor  which  is  represented  by 
the  ordinate  B,  and  then  a  slow  drop  of  pressure  along  the 
remaining  portion  of  the  small  pipe.  In  the  diagram  00,  the 
pump  and  motor  are  supposed  to  be  located  at  definite  points 
so  that  the  rise  of  pressure  in  the  pump  and  the  drop  of  pres- 
sure in  the  motor  are  represented  by  the  vertical  ordinates  A 
and  B. 

Figure  \*jb  represents  an  electric  generator  G  forcing  an  elec- 
tric current  through  a  small  conductor  and  through  a  distant 
electric  motor  M,  the  current  being  returned  to  the  generator 
through  a  very  large  conductor  of  negligible  resistance.  Choos- 
ing the  line  00  as  a  reference  axis,  the  electromotive  force  be- 
tween the  point  P  and  any  other  point  in  the  system  may  be 
represented  by  an  ordinate  measured  upwards  or  downwards 
from  the  reference  axis.  In  the  diagram  OO  the  generator  and 
motor  are  supposed  to  be  located  at  definite  points  so  that  the 
propelling  electromotive  force  of  the  generator  is  represented  by 
a  vertical  ordinate  A,  and  the  opposing  electromotive  force  of 
the  motor  is  represented  by  the  vertical  ordinate  B. 

When  one  has  chosen  a  reference  point,  like  P,  Fig.  ijb,  in  an 
electrical  system,  the  electromotive  force  between  that  point  and  any 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


other  point  in  the  system  is  called  the  electric  potential  at  the  other 
point.  —  Thus,  the  ordinates  /  and  /'    in  Fig.  i  jb  represent  the 


r 


B 


reference  axis 


O 


conductor 


Fig.  17b. 

values  of  the  electric  potential  at  the  two  points  q  and  q'  on 
the  wire  in  the  same  way  that  the  ordinates  p  and  p'  in  Fig. 
\ja  represent  the  values  of  the  hydrostatic  pressure  at  the  points 
q  and  q'  on  the  small  pipe,  that  is  to  say,  the  potential  at  a  point 
in  an  electrical  system  is  analogous  to  the  hydrostatic  pressure  at 
a  point  in  hydraulics  ;  and  the  electromotive  force  between  two 
points  in  an  electrical  system  which  by  definition  is  equal  to  the 
difference  of  potential  between  those  points,  is  analogous  to  the 
difference  of  pressure  between  two  points  in  a  hydraulic  system 

21.  Voltmeters*  and  ammeters.  —  Figure  iSa  shows  an  am- 
meter  A    arranged  to  measure  the  current  delivered  by  a  gener- 


main 


x  supply  mam 


supply  main 


Fig.   18a. 


Fig.   18b. 


*The  voltmeter  is  essentially  a  high-resistance  ammeter  except  in  the  case  of  the 
electrostatic  voltmeter  which  is  seldom  used.  Thus,  an  ammeter  gives  a  definite  de- 
flection with  a  certain  current  /  flowing  through  it,  and  the  electromotive  force  be- 


RESISTANCE    AND    ELECTROMOTIVE   FORCE.  43 

ator  G,  and  a  voltmeter  V  connected  so  as  to  indicate  the 
electromotive  force  between  the  terminals  of  the  generator. 
Figure  18$  shows  an  ammeter  A  and  a  voltmeter  V  arranged 
to  measure  the  power  delivered  to  a  lamp  L. 

An  ammeter  must  have  a  very  low  resistance  in  order  that  it 
may  not  obstruct  the  flow  of  current  in  a  circuit  in  which  it  is 
placed.  A  voltmeter  must  have  a  high  resistance  in  order  that 
it  may  not  take  sufficient  current  to  disturb  the  system  to  which 
it  is  connected.  Thus,  the  well-known  voltmeter  of  the  Weston 
Electric  Company  having  a  scale  ranging  from  zero  to  I  50  volts 
has  a  resistance  of  about  15,000  ohms,  so  that  it  takes  about 
o.oi  ampere  when  it  is  connected  to  a  i5O-volt  generator. 
When  an  ammeter  and  a  voltmeter  are  arranged  to  measure  the 
power  delivered  to  a  lamp,  as  shown  in  Fig.  186,  the  ammeter 
reading  should  be  taken  when  the  voltmeter  circuit  is  open  in 
order  that  the  ammeter  reading  may  indicate  the  true  current 
flowing  through  the  lamp.. 

22.  Polarization  *  of  the  voltaic  cell.  —  When  a  voltaic  cell 
delivers  current,  the  chemical  action  in  the  immediate  neighbor- 
hood of  the  electrodes  exhausts  the  electrolyte,  and  the  electro- 
motive force  of  the  cell  falls  off  greatly.  Thus,  the  ordinates  of 
the  curve  A  A  in  Fig.  19  represent  the  values  of  the  electro- 
motive force  of  a  dry  cell  after  it  has  been  delivering  a  fairly 
large  current  for  one  minute,  for  two  minutes,  for  three  minutes, 

tween  the  terminals  of  the  instrument  is  equal  to  AV,  where  J?  is  the  resistance  of 
the  instrument.  If  the  instrument  is  to  be  used  as  an  ammeter  the  position  of  the 
pointer  is  marked  with  the  number  which  gives  the  value  of  7  in  amperes,  if  the  in- 
strument is  to  be  used  as  a  voltmeter  the  position  of  the  pointer  is  marked  with  the 
number  which  gives  the  value  of  RI  in  volts. 

The  instrument  described  in  Art.  I  and  shown  in  Fig.  3  may  be  considered  to  be 
a  voltmeter  if  it  has  a  high  resistance. 

*  The  word  polarization  has  two  distinct  meanings  in  its  application  to  electrolysis. 
The  polarization  of  a  voltaic  cell  means  the  decrease  of  electromotive  force  of  the  cell 
due  chiefly  to  changes  of  concentration  of  the  electrolyte  in  the  neighborhood  of  the 
electrodes  of  the  cell  as  the  cell  delivers  current ;  and  the  polarization  of  an  electrode, 
as  this  term  is  generally  used  in  scientific  writings,  means  th«  total  electromotive  force 
between  the  electrode  and  the  electrolyte.  See  Practical  Physics,  Franklin,  Craw- 
ford and  MacNutt,  Vol.  II,  pages  136-147. 


44 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


and  so  on,  the  electromotive  force  being  measured  in  each  case 
on  open  circuit  (the  cell  being  disconnected  from  the  circuit  to 
which  it  delivers  current  and  connected  to  a  voltmeter  for  a 
moment  when  it  is  desired  to  read  its  electromotive  force). 


140 


f 


14       ID       1.8 
minutes 

Fig.  19. 


24:       26      28      3Q      3? 


When  a  voltaic  cell  has  been  polarized  by  delivering  current 
for  some  time,  its  electromotive  force  rises  slowly  when  it  is  left 
standing  on  open  circuit.  This  recovery  of  a  voltaic  cell  from 
polarization  is  due  chiefly  to  the  refreshing  of  the  electrolyte  in 
the  neighborhood  of  the  electrodes  by  the  slow  diffusion  of  the 
acid  or  salt  from  distant  portions  of  the  electrolyte  to  the  surfaces 
of  the  electrodes.  The  ordinates  of  the  curve  BB  in  Fig.  19 
show  the  increasing  values  of  the  electromotive  force  of  a  dry 
cell  standing  on  open  circuit  after  it  has  been  allowed  to  deliver 
current  for  some  time. 

BRANCHED  CIRCUITS. 

23.  Series  and  parallel  connections.  — When  two  portions  of  an 
electric  circuit  are  'so  connected  that  the  entire  current  in  the 
circuit  passes  through  both  portions,  the  portions  are  said  to  be 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  45 

connected  in  series.  When  two  portions  of  an  electrical  circuit 
are  so  connected  that  the  current  in  the  circuit  divides  and  part 
of  it  flows  through  each  portion,  the  portions  are  said  to  be  con- 
nected in  parallel.  Thus,  Fig.  20  shows  two  lamps  L  and  L' 


X          ft,          X 

t 9       T 


Fig.  20.  Fig.  21. 

connected  in  series,  and  Fig.  21  shows  two  lamps  connected  in 
parallel. 

The  ordinary  arc  lamps  which  are  used  to  light  city  streets  are 
connected  in  series,  and  the  entire  current  delivered  by  the  light- 
ing generator  flows  through  each  lamp.  On  the  other  hand,  if 
the  electromotive  force  of  the  generator  is,  say,  2,000  volts  and 
if  there  are  40  similar  *  lamps  in  series,  the  electromotive  force 
between  the  terminals  of  each  lamp  will  be  50  volts.  The  electro- 
motive force  of  a  generator  is  subdivided  among  a  number  of  lamps 
or  other  units  connected  in  series. 

The  ordinary  glow  lamps  which  are  used  for  house-lighting  are 
connected  in  parallel  between  copper  mains  which  lead  out  from 
the  terminals  of  the  generator,  and,  except  for  a  slight  drop  of 
electromotive  force  in  the  mains,  the  full  electromotive  force  of 
the  generator  acts  upon  each  lamp.  On  the  other  hand,  if  the 
generator  delivers,  say,  1,000  amperes  and  if  there  are  2,000  simi- 
lar *  lamps  connected  between  the  mains,  the  current  in  each  lamp 
will  be  one  half  ampere.  The  current  delivered  by  a  generator  is 
subdivided  among  a  number  of  lamps  or  other  units  connected  in 
parallel. 

Voltaic  cells  are  often  connected  in  series.  When  this  is  done 
the  electromotive  force  which  is  available  for  the  maintenance  of 
•current  is  equal  to  the  sum  of  the  electromotive  forces  of  the  indi- 

*  Having  the  same  resistance. 


46 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


vidual  cells.      Figure  22  is  a  top  view  of  three  dry  cells  connected 
in  series  and  delivering  current  to  a  circuit    R. 

A  number  of  voltaic  cells  of  the  same  kind  are  often  connected 
in  parallel.  When  this  is  done -the  total  current  delivered  by  the 
set  is  equal  to  the  sum  of  the  currents  delivered  by  the  individual 


R 


M/WVWWVWWWWW 


Fig.  22. 

cells,  and  the  electromotive  force  of  the  set  is  the  same  as  the 
electromotive  force  of  a  single  cell.  Figure  23  is  a  top  view  of 
three  dry  cells  connected  in  parallel  and  delivering  current  to  a 
circuit  R. 

Sometimes  it  is  desirable  to  connect  a  number  of  cells  in  groups, 
each  group  containing  a  number  of  cells  in  series,  and  to  connect 


NWWWWVWVWVWVK 

R 

Fig.  24. 

these  groups  of  cells  in  parallel.  Figure  24  is  a  top  view  showing 
two  groups  of  dry  cells  connected  in  parallel,  each  group  consist- 
ing of  four  cells  connected  in  series. 

24.  Discussion  of  the  division  of  current  in  two  branches  of 
a  circuit.  —  Figure  25  shows  a  battery  delivering  current  to  a 
circuit  which  branches  at  the  points  A  and  B.  Let  /  be  the 
current  in  the  main  circuit,  F  the  current  in  the  upper  branch, 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  47 

I"  the  current  in  the  lower  branch,  Rf  the  resistance  of  the 
upper  branch,  and  R"  the  resistance  of  the  lower  branch.  The 
product  R'F  is  the  electromotive  force  between  the  points  A 
and  B,  the  product  R" I"  is  also  equal  to  the  electromotive  force 
between  the  points  A  and  B,  and  therefore  we  have 

R'P=R"I"  (12)* 

The  current  in  the  main  part  of  the  circuit  is  equal  to  the  sum 
of  the  currents  in  the  various  branches  into  which  the  circuit 
divides.  Therefore  we  have  the  equation 

/=/'+/"  (13)* 

It  is  an  easy  matter  to  determine  the  values  of  /'  and  I" 
[with  the  help  of  equations  (12)  and  (13)]  in  terms  of  the  total 
current  /  and  the  resistances  R'  and  R"  of  the  respective 
branches.  It  is  important  to  note  that  a  a  definite  fractional  part 
of  the  total  current  flows  through  each  branch,  and  equation  (12) 
shows  that  the  currents  I'  and  I"  are  inversely  proportional  to  the 
resistances  R'  and  R"  respectively.  Thus,  if  Rf  is  nine  times 
as  large  as  R",  then  I"  is  nine  times  as  large  as  /',  so  that 
I"  must  be  equal  to  nine  tenths  of  7,  and  /'  must  be  equal  to 
one  tenth  of  /. 

25.  Combined  resistance  of  a  number  of  branches  of  a  circuit.  — 
(a)  The  combined  resistance  of  a  number  of  lamps  or  other  units 
connected  in  series  is  equal  to  the  sum  of  the  resistances  of  the 
individual  lamps.  (&)  The  combined  resistance  of  a  number  of 

*  Equations  (12)  and  (13)  express  two  principles  which  were  first  enunciated  by 
Kirchhoff  and  which  are  usually  called  KirchhofPs  laws,  as  follows  : 

(a)  Equation  (12)  may  be  written  J?'f  —  £"f"=o,  which  means  that  the 
sum  of  the  RI  drops  taken  in  a  chosen  direction  around  the  mesh  formed  by  the  two 
branches  of  the  circuit  is  equal  to  zero.  This  relation  is  true  of  a  mesh  of  any  net- 
work of  conductors.  If  one  side  of  the  mesh  contains  a  voltaic  cell  of  which  the 
electromotive  force  is  E,  then  the  sum  of  the  RI  drops  around  the  mesh  is  equal 
to  E. 

(6)  Equation  (13)  may  be  written  / — /'  —  /"^o,  which  means  that  the  sum 
of  the  currents  flowing  towards  one  of  the  branch  points  A  or  B  is  equal  to  zero. 
This  relation  may  be  generalized  as  follows :  The  sum  of  the  currents  flowing  towards 
a  branch  point  in  any  network  of  conductors  is  equal  to  zero. 


48  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

lamps  or  other  units  connected  in  parallel  is  equal  to  the  recipro- 
cal of  the  sum  of  the  reciprocals  of  the  respective  resistances. 
The  proposition  (a)  is  almost  self-evident.  Proposition  (ft)  may 
be  established  as  follows  :  Let  :-E  be  the  electromotive  force  be- 
tween the  points  A  and  B  where  the  circuit  divides  into  a 
number  of  branches.  Then,  according  to  Ohm's  Law,  we  have 


where  Rf,  R"  and  R"  '  are  the  resistances  of  the  respective 
branches,  and  /',  I"  and  I"  f  are  the  currents  flowing  in  the 
respective  branches. 

Let  /  bethetotalcurrentflowinginthecircuit(=//  +  ///-h////)- 
The  combined  resistance  of  the  branches  is  defined  as  the  resist- 
ance through  which  the  electromotive  force  E  between  the 
branch  points  would  be  able  to  force  the  total  current  /.  That 
is,  the  combined  resistance  is  defined  by  the  equation 


in  which  Rc  is  the  combined  resistance.  Adding  equations  (i), 
(ii)  and  (iii),  member  by  member,  and  substituting  EJRC  for 
/'  +/»  +  /"'  we  have 


whence 

R 


_L     _L       l 

R'  +  R"  +  R'" 


RESISTANCE   AND   ELECTROMOTIVE   FORCE.  49 

26.  Typical  problem  in  branched  circuits.  —  The  battery  in  Fig. 
2  5  has  an  electromotive  force  of  1 5  volts  ;  the  battery  and  the 
wires  which  connect  the  battery  to  the  points    A    and   B   have 
a  total  resistance  of  2  ohms ;  the  upper  branch  has  a  resistance 
of  3  ohms    (R!  =  3)    and  the  lower  branch  has  a  resistance  of  4 
ohms    (R"  =  4) ,  and  it  is  required   to   find  :  (a)  the  combined 
resistance  of  the  two  branches  and  total  resistance  of  the  circuit, 
(b)  the  total   current,  (c)  the  electromotive  force  between  the 
branch  points,  (d)  the  current  in  the  upper  branch,  and   (e)    the 
current  in  the  lower  branch. 

(a)  The  combined  resistance  of  the  two  branches  is  the  recip- 
rocal of  (^  +  ^),  or  J^2-  of  an  ohm.  Therefore  the  total  resistance 
of  the  circuit  through  which  the  battery  sends  current  is  3-J- 
ohms. 

(£)  The  total  current  is  found  by  dividing  the  electromotive 
force  of  the  battery  by  the  resistance  of  the  circuit,  which  gives 
4^  amperes. 

(c)  The  electromotive  force  between  the  branch  points  is  equal 
to  the  product  of  the  total  current  by  the  combined  resistance 
of  the  two  branches  or  to  4^  amperes  times  i|-  ohms,  which 
gives  6^|  volts. 

(d)  The  current  in  the  upper  branch  is  found  by  dividing  the 
electromotive  force  between  the  branch  points  by  the  resistance 
of  the  upper  branch,  which  gives  2^  amperes. 

(e)  The  current  in  the  lower  branch  is  found  by  dividing  the 
electromotive  force  between  the  branch  points  by  the  resistance 
of  the  lower  branch,  which  gives  1 1 1  amperes. 

27.  The  use  of  shunts  with  galvanometers  and  ammeters.  —  In 

the  use  of  a  galvanometer,  or  other  current-measuring  instrument, 
it  is  frequently  not  desirable  to  send  the  whole  of  the  current 
which  is  to  be  measured  through  the  instrument.  In  such  a  case 
a  definite  fractional  part  of  the  current  may  be  diverted  by  mak- 
ing the  instrument  one  of  two  branches  of  the  circuit,  as  shown 
in  Fig.  26a,  in  which  A  represents  the  galvanometer  or  ammeter 
5 


50  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

and   s    represents  the  auxiliary  branch.     This  auxiliary  branch 
is  called  a  shunt. 

Example.  —  A  galvanometer  (or  ammeter)  of  which  the  resist- 
ance is  R  ohms  is  shunted  by  a  resistance  of  -ft/99  ohms.  In 
this  case  99  times  as  much  current  flows  through  the  shunt  as 
through  the  galvanometer,  that  is,  y-J-^  of  the  total  current  flows 
through  the  galvanometer  and  -f^  of  the  total  current  flows 
through  the  shunt. 

28.  Use  of  voltmeter  multiplying  coils.  —  Suppose  one  has 
a  voltmeter  which  is  capable  of  indicating  the  value  of  any  elec- 
tromotive force  up  to  a  limit  of  10  volts  (more  than  10  volts 
throws  the  pointer  off  the  scale,  and  much  more  than  10  volts 
may  damage  the  instrument).  Let  R  be  the  resistance  of  the 


Fig.  26a.  Fig.  26b. 

instrument.  Let  an  auxiliary  resistance  equal  to  (n  —  \)R  be 
connected  in  series  with  the  instrument,  let  the  combination  be 
connected  to  an  electromotive  force  which  is  to  be  measured,  and 
let  E  be  the  reading  of  the  instrument ;  then  the  value  of  the 
electromotive  force  is  equal  to  nE.  This  is  evident  if  we  con- 
sider that  a  definite  deflection  on  the  voltmeter  means  a  definite 
current  flowing  through  the  instrument.  Let  this  current  be 
represented  by  /.  Then  the  electromotive  force  between  the 
terminals  of  the  instrument  is  RI  and  this  is  the  electromotive 
force  which  is  indicated  by  the  instrument  reading,  whereas 
the  electromotive  force  between  the  terminals  of  the  combination 
is  equal  to  the  product  of  the  current  times  the  resistance  of 
the  combination,  or  nRI. 

29.  Use  of  a  standard  shunt  and  a  millivoltmeter,  combined,  as 
an  ammeter.  —  A  millivoltmeter  is  a  voltmeter  for  reading  very 


RESISTANCE    AND    ELECTROMOTIVE   FORCE.  51 

small  electromotive  forces,  and  it  is  called  a  millivoltmeter  because 
its  scale  reading  indicates  the  value  of  an  electromotive  force  in 
millivolts  (one  millivolt  equals  one  one-thousandth  of  a  volt). 
The  current  /  to  be  measured  flows  through  a  known  low  resist- 
ance R,  and  the  electromotive  force  between  the  terminals  of 
this  resistance  is  measured  by  means  of  a  millivoltmeter  as  indi- 
cated in  Fig.  26$.  If  the  value  of  R  is  one  one-thousandth  of  an 
ohm,  then  the  reading  of  the  millivoltmeter  in  millivolts  is  the 
value  of  the  current  in  amperes.  If  the  value  of  R  is  one  one- 
hundredth  of  an  ohm,  then  the  reading  of  the  millivoltmeter  in 
millivolts  must  be  divided  by  10  to  give  the  value  of  the  current 
in  amperes.  If  the  value  of  R  is  one  tenth  of  an  ohm,  then 
the  reading  of  the  millivoltmeter  in  millivolts  must  be  divided  by 
100  to  give  the  value  of  the  current  in  amperes.  It  is  evident 
from  the, connections  shown  in  Fig.  26$  that  the  total  current  is 
equal  to  the  current  in  the  known  resistance  R  plus  the  current 
flowing  through  the  millivoltmeter  ;  but  inasmuch  as  the  resist- 
ance of  the  millivoltmeter  is  always  quite  large,  the  current  which 
flows  through  it  is  very  small  and  is  always  negligible  in  com- 
parison with  the  current  which  flows  through  R.  The  resistance 
R  in  Fig.  26$  forms  a  shunt  to  the  millivoltmeter  and  the  combi- 
nation exemplifies  the  matter  which  is  discussed  in  Art.  27. 

PROBLEMS. 

13.  A  current  of  0.5  ampere  flowing  through  a  glow  lamp 
generates  150  calories  of  heat  in  10  seconds,     (a)  Required  the 
resistance  of  the  lamp  in  ohms.     (&)  What  power  is  expended  in 
the  lamp?     Express  in  watts  and  in  horse-power.     Ans.  (a)  252 
ohms  ;  (^)  63  watts  or  0.0844  horse -power. 

14.  A   wire   having  a  resistance  of  250  ohms  is  coiled  in  a 
vessel  containing  2,000  grams  of  oil  of  which  the  specific  heat  is 
0.60.     The  vessel  itself  weighs  200  grams  and  its  specific  heat 
is  0.095.     A  current  of  1.5  amperes  is  passed  through  the  coil  of 
wire.      How  long  will  it  take  to  raise  the  temperature  of  the  oil 
and  the  vessel  one  centigrade  degree  ?     Ans.  9. 1 1  seconds. 


52  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

15.  The  field  coil  of  a  dynamo  contains  25  pounds  of  copper 
(specific  heat  0.094),  weight  of  cotton  insulation  negligible.     The 
resistance  of  the  coil  is   100  ohms,     (a)  At  what  rate  does  the 
temperature  of  the  coil   begin*  to   rise   when  a   current  of  0.5 
ampere  is  started  in  the  coil  ?     (&)  How  long  would  it  take  for 
the  temperature  of  the  coil  to   rise  to  20°  C.  if  no  heat  were 
given  off  from  the  coil  by  radiation?     Ans.   (a)  0.02345   centi- 
grade degree  per  second  ;  (ft)  14.2  minutes. 

16.  A  given  spool  wound  full  of  copper  wire  60  mils  in  diameter 
has  a  resistance  of  3.2  ohms.     An  exactly  similar  spool  is  wound 
full  of  copper  wire  1 20  mils  in  diameter.     What  is  its  resistance  ? 
Ans.  0.2  ohm. 

17.  What  is  the  resistance  at  20°  C.  of  2  miles  of  commercial 
copper  wire  300  mils  in  diameter?     Ans.  1.22  ohms. 

18.  What  is  the  resistance  at  20°  C.  of  one  mile  of  a  conductor 
consisting    of  seven  copper  wires    each   40   mils    in    diameter? 
Ans.  4.9  ohms. 

19.  Find  the  resistance  at  20°  C.  of  a  copper  conductor  100 
feet  long,  having  a  rectangular  section  0.5  x  0.25  inch.     Ans. 
0.00653  ohm. 

20.  A  sample  of  commercial  copper  wire  3  feet  long  and  120 
mils  in  diameter  is  found,  by  test,  to  have  at  the  same  tempera- 
ture a  resistance  equal  to  that  of  26.2  inches  of  pure  copper  wire 
100  mils  in  diameter.     Find  the  ratio  of  the  specific  resistance  of 
the  sample  to  the  specific  resistance  of  pure  copper.     Ans.  1.048. 

21.  What  is  the  resistance  at  20°  C.  of  a  steel  rail  30  feet  long 
weighing  900  pounds?     The  specific  gravity  of  the  steel  is  7.8. 
Ans.  0.000191  ohm. 

22.  What  is  the  resistance  at  20°  C.  of  an  iron  pipe  120  feet 
long  having  I  inch  inside  diameter,  and  ij  inches  outside  diam- 
eter ?     The  pipe  has  seven  joints,  and  each  joint  is  assumed  to 
have    the    resistance    of  one   foot    of  pipe.     Specific    resistance 
assumed  to  be  the  same  as  rail  steel.     Ans.  0.01536  ohm. 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  53 

23.  A  pure  copper  wire,  2,000  feet  long  weighs  125  pounds. 
What  is  its   resistance  at  20°  C.      How  will  its  resistance  be 
changed  by  doubling  the  length  without  changing  its  weight  ? 
The  specific  gravity  of  copper  is  8.9.     Ans.  i.oi  ohms. 

24.  The  specific  resistance  of  carbon  such  as  used  for  arc  lamps 
is  about  2,400  times  as  great  as  that  of  pure  copper.     Find  the 
watts  lost,  that  is,  find  Ri2,  in  the  two  carbons  of  an  arc  lamp,  8 
inches  of  each  carbon  being  in  circuit,  the  carbon  being  J  inch  in 
diameter,  and  the  current  passing  through  the  lamp  being  9.6 
amperes.     Ans.  12.3  watts. 

25.  A  column  of  a   1  5  per  cent,  solution  of  CuSO4,    I   meter 
long,  having  one  square  millimeter  section,  has  a  resistance  of 
260,000  ohms.     An  electrolytic  cell  of  this  solution  has  two  flat 
electrodes,  30  x  30  centimeters,  2.5  centimeters  apart.     Calculate 
the  current  due  to  2  volts  between  electrodes,  allowing  0.2  volt 
for  polarization.     Ans.  27.7  amperes. 

26.  A  copper  transmission  line  has  a  resistance  of  5  ohms  at 
20°  C.     What  is  its  resistance  at  90°  C.  ?     Ans.  6.297  ohms. 

Note.  —  The  difference  between  the  temperature  coefficient  of  resistance  of  a  metal 
expressed  as  a  fraction  of  its  resistance  at  o°  C.  and  its  temperature  coefficient 
expressed  as  a  fraction  of  its  resistance  at  any  other  temperature  not  greatly  different 
from  o°  C.  is  less  than  the  variations  of  the  temperature  coefficient  for  different 
samples  of  the  same  (commercial)  metal,  and  therefore  it  is  ridiculous  to  insist  on 
the  refined  calculations  which  grow  out  of  the  above-mentioned  difference.  The 
answers  to  all  temperature-resistance  problems  in  this  collection  are,  however,  found 
by  the  correct  (arithmetically  correct  0  method.  The  formula  is 


27.  A  wire  has  a  resistance  of  164.8  ohms  at  20°  C.  and  a 
resistance  of  186.2  ohms  at  70°  C.     What  is  the  mean  tempera- 
ture coefficient?     Ans.  0.002739. 

28.  The  field  coil  of  a  dynamo  has  a  resistance  of  42.6  ohms 
after  the  dynamo  has  stood  for  a  long  time  in  a  room  at  20°  C. 
After  several  hours'   running  the  resistance  of  the  coil  is   51.6 
ohms.     What  is  its  temperature?     Ans.  76.5°. 

29.  A  platinum  wire  has  254  ohms  resistance  at  o°  C.     When 


54          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

placed  in  a  furnace  its  resistance  is  1,630  ohms.     What  is  the 
temperature  of  the  furnace  ?     Ans.  1,531°. 

30.  A  platinum  wire  which  has  254  ohms  resistance  at 
o°  C.  has  a  resistance  of  81  tfhms  when  placed  in  a  bath  of 
liquid  air.  What  is  the  temperature  of  the  liquid  air  ?  Ans. 


. 

31.  A  glow  lamp  has  a  resistance  of  220  ohms  at  a  tempera- 

ture of  1,000°  C.  (a  bright  red  heat).  At  20°  C.  its  resistance  is 
277  ohms.  What  is  the  mean  temperature  coefficient  of  the 
carbon  filament?  Ans.  —  0.00021. 

32.  The  temperature  coefficient  of  a  given  metal  is  0.004  Per 
degree  centigrade  when  expressed  in  terms  of  the  resistance  of  the 
metal   at  o°    C.      Find  the  temperature   coefficient  per  degree 
Fahrenheit  expressed  in  terms  of  the  resistance  at  o°  F.     Ans. 
0.00239  per  degree  F. 

Note.  —  Assume  a  wire  of  the  given  metal  of  which  the  resistance  at  o°  C.  is  one 
ohm  and  calculate  its  resistance  R  at  —  17.78°  C.  (equals  o°  F.).  The  tempera- 
ture coefficient  per  degree  centigrade  expressed  in  terms  of  the  resistance  at  —  17.78° 
C.  is  greater  than  the  temperature  coefficient  per  degree  centigrade  expressed  in  terms 
of  the  resistance  at  o°  C.  in  the  ratio  of  R  to  unity  and  this  result  must  be  divided 
by  1.8  to  get  the  coefficient  per  degree  Fahrenheit  in  terms  of  the  resistance  at  o°  F. 

33.  Practically  all  of  the  energy  of  the  chemical  action  which 
takes  place  in  the  gravity  cells  goes  to  maintain  the  current  pro- 
duced by  the  cell.     When  one  gram  of  powdered  zinc  is  stirred 
into  a  solution  of  copper  sulphate  756  calories  of  heat  are  gener- 
ated.     Calculate  the   electromotive  force  of  the    Daniell    cell. 
Ans.  1.07  volts. 

JVbte.  —  Assume  the  current  of  one  ampere  and  find  the  fraction  of  a  gram  z  of  zinc 
which  would  be  deposited  by  this  current  per  second.  This  is  the  amount  of  zinc 
which  is  consumed  per  second  by  voltaic  action.  Find  the  number  of  calories  of  heat 
represented  by  the  reaction  of  z  grams  of  zinc  with  copper  sulphate,  and  reduce  this 
result  to  joules.  We  thus  find  the  number  of  joules  per  second  developed  by  the  vol- 
taic action  which  is  produced  when  one  ampere  flows  through  the  cell  and  this  is 
equal  to  the  desired  electromotive  force  in  volts. 

34.  A  fan  blower  develops  between  its  inlet  and  outlet  a  pres- 
sure-difference of  three  fourths  pound  per  square  inch.     When 
the  outlet  is  open  the  fan  delivers  20  cubic  feet  of  air  per  second. 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  55 

At  what  rate  does  the  fan  do  work  in  delivering  this  air  ?     Ans. 
2,160  foot  pounds  per  second. 

Note.  —  Reduce  the  pressure-difference  to  pounds  per  square  foot  and  then  the 
unit  in  terms  of  which  the  result  is  expressed  will  be  at  once  evident. 

35.  When  a  certain  electric  generator  is  giving  out  no  current 
it  takes    1.75    horse-power  to  drive  it.     When  the  generator  de- 
livers a  current  of  150  amperes  it  takes  25  hbrse-power  to  drive 
it.     Assuming  that  the  increased  power  is  all  used  in  the  main- 
tenance of  the   1 50  amperes  of  current,  find   the  electromotive 
force  of  the  generator.     Ans.  1 1 5:7  volts. 

36.  An  incandescent  lamp  takes  0.6  ampere  when  the  electro- 
motive force  between  its  terminals  is  1 10  volts.     Find  the  power 
delivered  to  the  lamp  in  watts  and  in   horse-power.     Ans.   66 
watts  or  0.0884  horse-power. 

37.  In  the  electrolytic  refining  of  copper  an  electromotive  force 
of  0.3  of  a  volt  suffices  to  send  the  current  through  the  electro- 
lytic cell  in  which  the  pure  copper  is  deposited.      Calculate  the 
number   of  kilowatt-hours   required    to   deposit  a  ton   of  pure 
copper.     Ans.  230  kilowatt-hours. 

Note.  —  See  data  on  pages  23  and  24. 

38.  In  the  electrolytic  manufacture  of  aluminum  by  electrolysis 
an  electromotive  force  of  5.5  volts  suffices  to  send  the  current 
through  the  electrolytic  cell  in  which  the  metallic  aluminum  is 
deposited.     Find   the  number  of  kilowatt-hours  required  for  the 
production  of  one  ton  of  aluminum.     Ans.  14,810  kilowatt-hours. 

Note.  —  One   kilowatt   continuously  for  one   year  costs   from  $20  to  $40  when 
developed  on  a  large  scale  by  water  power.     See  data  on  pages  23  and  24. 

39.  When  electrical  energy  costs    15   cents  per  kilowatt-hour 
how  much  does  it  cost  to  operate,  for   10  hours,  a  glow  lamp 
which  takes  \  an  ampere  from  no- volt  mains?     Ans.  8J  cents. 

40.  An  electric  motor  which  delivers  5  horse-power  at  its  belt 
has  an  efficiency  of  85  per  cent.     This  motor  is  supplied  with  cur- 
rent from   no-volt  mains.     What  current  does  it  take?     Ans. 
39.89  amperes. 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


41.  A  fine  copper  wire  wound  in  one  layer  upon  a  pane  of  glass 
is  submerged  in  an  oil-bath  and  a  measured  current  /  is  allowed 
to  flow  through  the  wire  causing  the  temperature  of  the  bath  to 
rise  slowly.  A  voltmeter  is  connected  across  the  terminals  of  the 
coil  of  copper  wire  and  simultaneous  readings  of  current  /  in 
the  coil,  electromotive  force  E  across  the  terminals  of  the  coil 
and  temperature  T  of  the  bath  were  taken  as  follows  : 


r 

E 

7 

T 

E 

7 

20°  C. 

55.80  volts. 

4.  74  amp. 

55°  C. 

56.00  volts. 

4.  2  1  amp. 

25 

55.80 

4-65 

60 

56.15 

4.15 

3° 

55.80 

4-50 

65 

56.20 

4-05 

35 

55.80 

4.48 

70 

56.25 

4.025 

40 

55.85 

4.41 

75 

56.30 

3-96 

45 

55-95 

4-35 

80 

56.35 

3.91 

50 

55-95 

4.28 

85 

56.40 

3-85 

90 

56.40 

3-79 

Calculate  the  resistance  of  the  wire  at  each  observed  temperature, 
and  plot  a  curve  of  which  the  abscissas  represent  observed  tem- 
peratures and  of  which  the  ordinates  represent  the  calculated 
values  of  the  resistance  of  the  wire. 

42.  An  electrolytic  cell,  consisting  of  a  one  per  cent,  solution 
of  sulphuric  acid  between  lead  electrodes,  was  connected  to 
supply  mains,  and  the  following  values  of  current  /  flowing 
through  the  cell,  electromotive  force  E  between  the  electrodes, 
and  temperature  T  of  the  solution  were  observed.  Calculate 
the  resistance  of  the  cell  at  each  temperature,  and  plot  a  curve 
of  which  the  abscissas  represent  temperatures,  and  of  which  the 
ordinates  represent  the  corresponding  calculated  resistances  of 
the  cell. 


7 

E 

T 

7 

E 

T 

3.3  amp. 
3-73 
4-35 
4.68 

56.  1  volts. 
54-o 
54  6 
53-0 

22.95°  C. 

30 
40 
50 

5.  06  amp. 
5-47 
5-75 
6.00 

52.  6  volts. 
53-1 
52.9 
52.5 

60°  C. 

90 

Note.  —  The  resistance  of  the  cell  in  this  problem  is  to  be  calculated  by  means  of 
Ohm's  Law.  Ohm's  Law,  however,  is  not  strictly  applicable  in  this  case,  because  a 
portion  of  the  work  which  is  done  on  the  cell  is  used  to  produce  chemical  action, 
whereas  Ohm's  Law  is  true  only  in  case  all  the  work  delivered  to  a  circuit  is  spent  in 


RESISTANCE   AND    ELECTROMOTIVE   FORCE.  57 

heating  the  circuit  in  accordance  with  Joule's  Law.  This  matter  may  be  stated  in 
another  way,  as  follows  :  A  certain  portion  of  the  observed  electromotive  force  is 
used  to  produce  chemical  action,  and  the  remainder  is  used  to  overcome  the  re- 
sistance of  the  electrolyte  in  accordance  with  Ohm's  Law.  The  portion  of  the 
electromotive  force  which  is  used  to  produce  chemical  action  is  about  2  or  2^ 
volts,  so  that  but  little  error  is  introduced  by  ignoring  this  effect  and  assuming  that 
the  whole  of  the  electromotive  force  is  used  to  overcome  resistance  in  accordance 
with  Ohm's  Law. 

43.  A  coil  of  which  the  resistance  is  to  be  determined  is  con- 
nected in  series  with  an  ammeter  across  iio-volt  mains,  and  the 
current  is  observed  to  be  26  amperes.     What  is  the  resistance  of 
the  coil?     Ans.  4.23  ohms. 

44.  A  wire  of  which  the  resistance  is  1 50  ohms  is  connected 
to  the  terminals  of  a  1 1  o-volt  dynamo,  and  a  point  on  this  wire 
is  grounded,  the  resistance  between  the  positive  terminal  of  the 
dynamo  and  the  grounded  point  being  60  ohms.     Choosing  the 
ground  as  the  region  of  zero  potential,  find  the  potential  of  each 
terminal  of  the  dynamo.     Ans.  The  potential  of  the   positive 
terminal  is  4-  40  volts,  and  the  potential  of  the  negative  terminal 
is  —  60  volts. 

45.  A  voltaic  cell  of  which  the  electromotive  force  is  1.07  volts 
and  the  resistance  is  2. 1   ohms  is  connected  to  a  coil  of  5  ohms 
resistance,     (a)  What  current  is  produced  ?    (&)  What  is  the  elec- 
tromotive force  drop  in  the  cell  ?     (c)  What  is  the  electromotive 
force  between  the  terminals  of  the  cell  ?      Ans.  (a)  o.  1 5  ampere, 
(b)  0.32  volt,  (c)  0.75  volt. 

46.  A  storage  battery  consisting  of  54  cells  connected  in  series 
has  a  resistance  of  0.0002  ohm  per  cell,  and  an  electromotive 
force  per  cell  which  ranges  from  2  volts  at  the  beginning  to  1.85 
volts  at  the  end  of  the  discharge.     The  battery  supplies  current 
to  100  glow  lamps  (each  having  220  ohms  resistance)  connected 
in  parallel  between  copper  wires  0.325  inch  in  diameter  at  a  dis- 
tance of  200  feet  from  the  battery.      Find  the  electromotive  force 
between  the  terminals  of  the  group  of  lamps  at  the  beginning 
and  at  the  end  of  the  discharge  of  the  storage  battery.     Ans. 
105.6  volts  and  97.7  volts. 


58  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

47.  The  electromotive  force  of  a  battery  is  1 5  volts  (measured 
on  open  circuit).     The  battery  terminals  are  connected  by  a  wire, 
when  it  is  observed  that  a  current  of  1.5  amperes  is  produced 
and  the  electromotive-  force  between  the  battery  terminals  is  9 
volts.      Find  the  resistance  of  the  wire  and  the  apparent  resistance 
of  the  battery.     Ans.   6  ohms  and  4  ohms. 

Note.  —  When  a  voltaic  cell  is  called  upon  to  give  current,  the  terminal  voltage  of 
the  cell  falls  off,  not  only  on  account  of  the  ri  drop  in  the  cell,  but  also  on  account 
of  what  is  called  polarization.  This  problem  is  to  be  solved  on  the  assumption  that 
the  whole  of  the  decrease  in  terminal  voltage  is  due  to  ri  drop.  The  value  of  the 
resistance  as  calculated  on  this  assumption  is  greater  than  the  true  resistance  of  the 
battery. 

48.  Find  the  total  electromotive  force  that  must  be  induced  in 
a  dynamo  armature  to  send  a  charging  current  of  100  amperes 
through  a  storage  battery  consisting  of  54  cells   connected  in 
series.      Each  cell  has  an  average  counter  electromotive  force  of 
2.3  volts,  the  resistance  of  each  cell  is  0.0004  ohm,  the  resistance 
of  the  dynamo  armature  is  0.02  ohm,  and  the  resistance  of  the 
leads  is  0.03  ohm.     Ans.    131.36  volts. 

49.  A  dynamo    having  an  electromotive  force  of   115   volts 
between  its  terminals  delivers  200  amperes  to  a  group  of  glow 
lamps  1 ,000  feet  distant  from  the  generator.      Find  :    (a]  the  size 
of  copper  wire  for  the  mains  in  order  that  95  per  cent,  of  the 
power  output  of  the   generator  may  be  delivered   to  the  lamps ; 
(<£)   the  electromotive  force   between  the  mains  at  the    lamps. 
Ans.  (a)  792,300  circular  mils;  (b)  109.25  volts. 

50.  What  size  of  copper  wire  is  required  to  deliver  current  at 
no  volts  to  a  i o-horse-power  motor  of  85  per  cent,  efficiency, 
the  motor  being  2,000  feet  from  the  generator,  and  the  electro- 
motive force  between  the  generator  terminals  being   125  volts. 
Ans.   221,200  circular  mils. 

51.  A  motor  receiving  100  kilowatts  of  power  is  at  a  distance 
of  1 5  miles  from  the  generator.     Line  wires  200  mils  in  diameter 
are  to  be  used.     The  line  loss  is  to  be  10  per  cent,  of  the  gener- 
ator output.     Find  :    (a)  the  current ;  (fr)  the  voltage  at  the  gen- 
erator, and  (r)  the  voltage  at  the  motor.     Ans.  (a)  16.44  amperes ; 
(b)  6,763  volts  ;  (c)  6,087  volts- 


RESISTANCE   AND    ELECTROMOTIVE    FORCE. 


59 


main  A 


main  B 


Note.  —  High-voltage  direct-current  power  transmission  is  not  much  used  in  Amer. 
ican  practice. 

52.  A  motor  using   100  kilowatts  of  power  is  10  miles  from 
the  generator.      Line  wires  200  mils  in  diameter  are  to  be  used. 
What  electromotive  force  is  required   at  the  generator  in  order 
that  the  line  loss  may  be  only  5   per  cent,  of  the  output  of  the 
generator?     Ans.  7,602  volts. 

53.  A    direct-reading    voltmeter    V,     Fig.  27,    having    16,000 
ohms  resistance,  is  connected  from  main  A   to  earth.     The  volt- 
meter gives  a  reading  of  2.6  volts  and  the  electromotive  force 
between  the  mains  is  1 10  volts. 

Find  the  insulation  resistance 
between  main  B  and  the  earth 
on  the  assumption  that  the  in- 
sulation resistance  of  main  A 
is  :  (a)  infinite  ;  (fr)  the  same  as 
that  of  main  B ;  (c)  one  tenth 
of  that  of  main  B.  Ans.  (a) 
660,900  ohms ;  (fr)  644,900 
ohms;  (c)  500,900  ohms. 

54.  Three  resistances    A,  B   and    C  of  which  the  values  are 
500  ohms,  200  ohms  and  1.2  ohms,  respectively,  are  connected 
to  a  battery  of  negligible  resistance,  the  electromotive  force  of  the 
battery  being  2   volts.     The  connections  are  made  so  that  the 
whole  current  produced  by  the  battery  flows  through    A,    then 
divides  and  passes  through    B   and    C  in  parallel,  and  returns  to 
the  battery.     Calculate  the  total  resistance  of  the  circuit,  the  total 
current,  the  current  in    B,   the  current  in    C,    the  electromotive 
force  between  the  terminals  of  A,    and  the  electromotive  force 
between    the   terminals    of    B    (or     C).      Ans.    501.19    ohms, 
0.00399   ampere,   0.000024   ampere,  0.003966    ampere,   1.995 
volts,  0.005  volt,  in  order. 

55.  Three  resistances  of  4,  4  and  2  ohms  respectively  are  con- 
nected in  parallel ;  and  two  resistances  of  6  and  3  ohms  in  parallel. 
The  first  combination  is  connected  in  series  with  the  second,  and 


earth 


Fig.  27. 


60  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

with  a  battery  of  three  volts  electromotive  force  and  negligible 
resistance.  What  is  the  current  in  the  2-ohm  and  3-ohm  resist- 
ances ?  Ans.  0.5  ampere,  0.66  ampere. 

56.  Six  voltaic  cells,  each  having  a  resistance  of  2  ohms,  are 
connected  to  a  coil  of  which  the  resistance  is  5  ohms.     What  is 
the  total  resistance  of  the  circuit :  (a)  When  the  6  cells  are  con- 
nected in  series,  (fr)  when  the  6  cells  are  connected  2  in  parallel 
by  3  in  series ;     (c)  when  the   6  cells  are  connected  3  in  parallel 
by  2  in  series  ;  and  (d)  when  the  6  cells  are  connected  in  paral- 
lel ?     Ans.  (a)  17  ohms,   (b)    8  ohms,  (c)  6J    ohms,  and  (d)  5^- 
ohms. 

The  electromotive  force  of  each  cell  is  1.6  volts.  What  current 
is  produced  in  the  coil  in  each  of  the  above  cases  ?  Ans.  (a)  0.57 
ampere,  (fr)  0.6  ampere,  (c}o.$  ampere,  (d)  0.3  ampere. 

Note.  —  When  n  voltaic  cells  are  connected  in  series,  their  combined  electromotive 
force  is  n£,  where  E  is  the  electromotive  force  of  one  cell. 

57.  A  direct- reading  ammeter  has  a  resistance  of  0.05  ohm. 
The  instrument  is  provided  with  a  shunt  so  that  the  total  current 
passing  through  the  instrument  and  shunt  is  10  times  the  ammeter 
reading.     What  is  the  resistance  of  the  shunt?     Would  it  be 
practicable  to  construct  such  a  shunt,  measure  its  resistance  by  a 
Wheatstone's  bridge,  and  connect  it  to  the  ammeter  terminals  ? 
If  not,  how  could  such  a  shunt  be  accurately  adjusted  ?     Ans. 
0.00556  ohm. 

58.  A  millivoltmeter   has   a  resistance  of  15.4  ohms.     What 
resistance  must  be  connected  in  series  with  the  instrument  so  that 
the  scale  reading  may  give  volts  instead  of  millivolts  ?     Ans. 
15,384.6  ohms. 

59.  The  scale  of  a  direct-reading  millivoltmeter  has   100  divis- 
ions, each  division  corresponding  to  one  millivolt  between  the 
terminals  of  the  instrument.     This  instrument  is  connected  to  the 
terminals  of  a  low  resistance  shunt  and  each  division  of  the  scale 
corresponds  to  0.25  ampere  in  the  shunt.     What  is  the  resistance 
of  the  shunt  ?     Ans.  0.004  ohm. 


CHAPTER    III. 
THE   MAGNETISM   OF   IRON. 

30.  Ferromagnetism  and  electromagnetism.  —  There  are  two 
distinct  groups  of  magnetic  phenomena,  namely,  (a)  the  phe- 
nomena of  ferromagnetism,  that  is  to  say,  the  phenomena  which 
are  associated  with  magnetized  iron  and  steel,  and  (&)  the  phe- 
nomena of  electromagnetism,  that  is  to  say,  the  magnetic  phe- 
nomena which  are  exhibited  by  the  electric  current  in  the  ab- 
sence of  iron  and  steel.  In  developing  the  subject  of  magnetism 
it  is  necessary  to  study  ferromagnetism  first  because  the  phe- 
nomena of  ferromagnetism  are  much  more  familiar  than  the 
phenomena  of  electromagnetism ;  in  fact,  the  phenomena  of 
electromagnetism  are  comparatively  obscure,  and,  in  many  cases, 
almost  imperceptible,  except  when  they  are  enhanced  by  the 
presence  of  iron.  Thus,  a  dynamo  or  an  induction  coil  would 
operate  if  all  its  iron  parts  were  removed,  but  the  effects  pro- 
duced would  be  so  slight  as  to  be  almost  imperceptible.  Practi- 
cally, therefore,  the  phenomena  of  ferromagnetism  and  the  phe- 
nomena of  electromagnetism  are  inextricably  associated  with  each 
other.  In  the  rational  study  of  magnetism,  however,  a  considera- 
tion of  the  phenomena  of  the  magnetism  of  iron  leads  to  the  all- 
important  conception  of  the  magnetic  field,  and  the  subject  of 
electromagnetism  is  then  developed  on  the  basis  of  this  concep- 
tion as  exemplified  in  Chapters  IV,  V,  and  VI. 

The  magnet.  —  The  name  magnet  was  originally  applied  to  the 
lodestone,  a  mineral  composed  of  iron  oxide,  which,  in  its  native 
state,  possesses  the  power  of  attracting  iron. 

The  electromagnet.  —  One  aspect  of  the  magnetic  effect  of  the 
electric  current,  as  described  in  Art.  I  and  as  shown  in  Fig.  2,  is 
that  an  iron  rod  which  is  wound  with  an  insulated  wire  becomes 
a  magnet  when  an  electric  current  is  sent  through  the  wire. 

61 


62  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Such  an  iron  rod  with  its  winding  of  wire  is  called  an  electro- 
magnet. The  iron  rod  is  called  the  core,  the  coil  of  wire  is  called 
the  winding,  and  the  electric  current  which  flows  through  the 
coil  is  called  the  exciting  current'^  the  electromagnet. 

The  permanent  magnet.  —  When  the  core  of  an  electromagnet 
is  made  of  soft  iron  it  loses  *  its  magnetism  very  quickly  and 
almost  completely  when  the  exciting  current  ceases  to  flow. 
When  the  core  of  an  electromagnet  is  made  of  hardened  steel, 
however,  it  retains  its  magnetized  condition  very  persistently  after 
the  exciting  current  has  ceased  to  flow,  and  of  course 
such  a  bar  of  magnetized  steel  may  be  removed  from 
the  magnetizing  winding.  A  steel  bar  magnetized  in 
this  way  is  called  a  permanent  magnet.  A  perma- 
nent magnet,  so-called,  loses  its  magnetism  more  or 
less  .rapidly  when  it  is  subjected  to  mechanical  shocks 
or  temperature  changes. 

Poles  of  a  magnet.  Compass.  Naming  the  poles. — 
Certain  parts  only  of  a  magnet  possess  the  power  of 
attracting  iron.  These  parts  of  a  magnet  are  called  its 
poles.  The  poles  of  a  bar  magnet,  for  example,  are 
usually  at  its  ends.  Thus,  Fig.  28  shows  a  bar  mag- 
net with  iron  filings  clinging  to  its  ends.  A  horizontal  magnet 
which  is  free  to  turn  about  a  vertical  axis  places  itself,  at  most 
places  on  the  earth  approximately  north  and  south.  This  beha- 
vior of  a  magnet  is  exemplified  in  the  ordinary  magnetic  com- 
pass which  consists  of  a  pivoted  magnet  playing  over  a  divided 
circle.  The  terms  magnetic  north,  magnetic  east,  etc.,  are  occa- 
sionally used  in  referring  to  the  cardinal  points  of  the  compass 
as  indicated  by  the  compass  needle. 

The  north  pointing  pole  of  a  magnet  is  called  its  north  pole, 
and  the  south  pointing  pole  of  a  magnet  is  called  its  south  pole. 
Mutual  force  action  of  two  magnets.  — When  a  magnet  is  sud- 
denly brought  near  to  a  compass,  the  compass  needle  is  set  more 

*  Except  when  the  core  is  long  and  slim,  or  when  the  core  is  part  of  a  complete 
iron  circuit. 


THE   MAGNETISM   OF   IRON. 


or  less  violently  into  motion  (coming  quickly  to  rest)  because  of 
the  force  which  the  magnet  exerts  on  the  compass  needle.  In 
general,  any  two  adjacent  magnets  exert  forces  on  each  other, 
and  this  mutual  force  action  is  always  resolvable  into  four  parts, 
namely,  the  force  with  which  each  of  the  poles  of  one  magnet 
acts  upon  each  of  the  poles  of  the  other  magnet.  The  north  pole 
of  each  magnet  attracts  the  south  pole  of  the  other  magnet,  the 
north  poles  of  both  magnets  repel  each  other,  and  the  south 
poles  of  both  magnets  repel  each  other.  Unlike  magnetic  poles 
attract  each  other,  like  magnetic  poles  repel  each  other. 

31.  Distributed  and  concentrated  magnet  poles.  —  The  poles  of 
a  magnet,  that  is,  the  seats  of  the  attracting  and  repelling  forces 
above  described,  are  distributed  over  considerable  portions  of  the 
bar,  generally  the  end  portions.  This  is  especially  the  case  with 
short  thick  bars.  In  the  case  of  long  slim  magnets,  however,  the 
poles  are  ordinarily  more  nearly  concentrated  at  the  ends  of  the 
bar.  In  the  first  case  we  have  what  are  called  distributed  poles, 
and  in  the  second  case  we  have  what  are  called  concentrated  poles. 
The  laws  of  attraction  and  repulsion  of  magnets  are  quite  simple 
for  long  slim  magnets  with  concentrated  poles,  and  the  ideal  slim 
magnet  with  concentrated  poles  will  be  made  use  of  in  the  follow- 
ing development  of  the  fundamental  ideas  relating  to  the  mag- 
netism of  iron  and  to  the  magnetic  action  of  the  electric  current. 


N  N     N  N     N  N     N  N 


32.  Strength  of  a  magnet  pole.  —  The  poles  of  a  magnet  may 
attract  iron  with  greater  or  less  force 
according  to  the  size  of  the  magnet 
and  according  to  the  thoroughness 
with  which  the  magnet  has  been 
magnetized.  The  poles  of  a  magnet 
are  said  to  be  strong  when  they  at- 
tract iron  or  steel  with  a  relatively 
great  force.  Consider  a  pair  of  long 
slim  magnets  a,  Fig.  29,  another  pair  b,  another  pair  r,  another 
pair  d,  and  so  on,  the  two  magnets  of  each  pair  being  exactly 


a 

6 

c 

d 

s  s 


s  s    s  s 


Fig.  29. 


64          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

alike.  It  is  possible  to  select,  from  such  a  set,  two  similar  magnets 
of  which  the  two  north  poles  ,  for  example,  repel  each  other  with  a 
force  of  one  dyne  when  they  are  one  centimeter  apart.  Each  of 
the  poles  is  then  said'  to  be  of  one  unit  strength,  and  the  strength 
m  of  any  other  pole  is  equal  to  the  force  in  dynes  with  which  this 
other  pole  is  acted  upon  by  a  unit  pole  at  a  distance  of  one  centi- 
meter. The  force  with  which  two  poles,  of  strengths  m'  and  m"  , 
respectively,  attract  each  other  when  they  are  at  a  distance  of  one 
centimeter  apart  is  m'm"  dynes.  This  is  evident  when  we  con- 
sider that  each  of  the  m'  unit  poles,  which  may  be  thought  of  as 
being  combined  to  give  the  pole  m*  ',  attracts  each  of  the  m"  unit 
poles  which  may  be  thought  of  as  being  combined  to  give  the 
pole  m"  ,  with  a  force  of  one  dyne. 

33.  Coulomb's  law.  —  The  force  of  attraction  or  repulsion  of 
two  magnet  poles  is  inversely  proportional  to  the  square  of  the 
distance  between  them.  This  fact  was  discovered  in  1800  by 
Coulomb,  who  measured  the  force  of  attraction  of  two  magnet 
poles  at  different  distances  apart  and  found  the  force  to  vary 
inversely  with  the  square  of  the  distance.  A  long  slim  magnet 
was  suspended  horizontally  by  a  wire,  thus  forming  a  torsion 
pendulum.  One  of  the  poles  of  another  long  slim  magnet  was 
brought  near  to  one  of  the  poles  of  the  suspended  magnet,  the 
force  action  between  the  two  poles  produced  a  twist  in  the 
suspending  wire,  and  the  value  of  the  force  was  determined  from 
the  observed  amount  of  twist. 

Complete  expression  for  the  force  of  attraction  of  two  magnet 
poles.  —  According  to  the  previous  article,  two  poles  attract  or 
repel  each  other  with  a  force  of  m'm"  dynes  when  they  are  one 
centimeter  apart,  therefore,  according  to  Coulomb's  Law,  the 
poles  attract  or  repel  each  other  with  a  force  of  m'm"  fr2  dynes 
when  they  are  r  centimeters  apart  ;  that  is, 

m'm" 


in  which   m'  and   m"  are  the  respective  strengths  of  the  two 


THE   MAGNETISM   OF   IRON.  65 

magnet  poles,  r  is  their  distance  apart  in  centimeters,  and  F  is 
the  force  in  dynes  with  which  they  attract  or  repel  each  other. 

Algebraic  sign  of  magnet  pole.  —  The  poles  mr  and  m"  are 
alike  in  sign  when  both  are  north  poles  or  when  both  are  south 
poles.  On  the  other  hand,  m'  and  m"  are  unlike  in  sign 
when  one  is  a  north  pole  and  the  other  is  a  south  pole.  It  is 
customary  to  consider  a  north  pole  as  positive  and  the  south 
pole  as  negative.  The  force  in  equation  (15)  is  considered  as 
positive  when  it  is  a  repulsion. 

Two  poles  of  a  magnet  always  equal  in  strength  and  opposite  in 
sign. — The  behavior  of  a  magnet  in  what  is  called  a  uniform 
magnetic  field,  as  described  in  Art.  41,  shows  that  the  poles 
of  a  magnet  are  always  equal  in  strength  and  opposite  in  sign. 
A  bar  of  steel  may  be  irregularly  magnetized  so  as  to  have  one 
or  more  north  poles  and  one  or  more  south  poles,  but  the  sum 
total  of  the  north  polarity  is  equal  to  the  sum  total  of  the  south 
polarity.  When  a  magnet  is  broken  in  two,  each  piece  is  found 
to  be  a  complete  magnet  with  a  north  pole  and  a  south  pole. 
//  is  often  convenient,  nevertheless,  to  speak  of  an  isolated  magnet 
pole,  meaning  one  pole  of  a  very  long  magnet,  the  other  pole  being 
so  far  away  as  to  be  negligible  in  its  effects. 

34.  Magnetic  figures.  The  magnetic  field.  —  When  iron  filings 
.are  dusted  upon  a  pane  of  glass  which  is  placed  over  a  magnet, 
the  filings  tend  to  arrange  themselves  in  regular  filaments.  Slight 
tapping  of  the  glass  facilitates  the  arrangement  of  the  filings. 
Figure  30  is  a  photographic  reproduction  of  a  magnetic  figure 
obtained  in  this  way.  This  magnetic  figure  conveys  the  idea  that 
.something  emanates  from  one  end  of  the  magnet,  traverses  the 
surrounding  region  in  beautifully  curved  lines,  and  enters  the 
other  end  of  the  magnet.  In  fact,  the  entire  region  surrounding 
a  magnet  is  in  a  peculiar  physical  condition  as  is  shown  by  the 
behavior  of  a  compass  needle  when  the  compass  is  brought  into 
the  neighborhood  of  a  large  magnet. 

Wherever  a  compass  may  be  placed  in  the  neighborhood  of  a 
magnet,  the  compass  needle  points  in  a  definite  direction,  the 
6 


66 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


same  direction,  indeed,  as  would  be  taken  by  filaments  of  iron 
filings  at  that  place.  Any  region  in  which  a  compass  needle 
tends  to  point  in  a  definite  direction  is  called  a  magnetic  field,  and 


Fig.  30. 

the  direction  of  the  compass  needle  (arrow-head  thought  of  as  being 
at  the  north-pointing  end  of  the  needle)  is  the  direction  of  the  field 
at  the  place  where  the  compass  is  located. 

The  filaments  of  iron  filings  in  a  magnetic  figure  shown  in 
Fig.  30  indicate  the  trend  of  what  are  called  the  lines  of  force  of 
the  magnetic  field.  A  line  of  force  is  at  each  point  in  the  direc- 
tion of  the  field  at  that  point. 

Example.  —  The  fine  parallel  lines  in  Fig.  3 1   represent  the 

lines  of  force  of  a  magnetic 
field  in  which  a  bar  magnet 
NS  is  placed,  and  the  heavy 
arrows  FF  represent  the  forces 
with  which  the  field  acts  on  the 
two  poles  of  the  magnet  tend- 
ing to  turn  it  into  the  direc- 


Fig.  31. 


tion  of  the  field. 


35.  Intensity  of  a  magnetic  field  at  a  point.  —  A  magnetic  field 
has  been  defined  as  a  region  in  which  a  compass  needle  tends  to 


THE   MAGNETISM   OF   IRON.  67 

/ 

point  in  a  definite  direction,  and  the  tendency  of  the  needle  to 
point  in  a  definite  direction  is  due  to  the  fact  that  equal  *  and  op- 
posite forces  FFt  Fig.  31,  are  exerted  on  the  two  poles  of  a 
magnet  by  the  magnetic  field,  that  is  to  say,  a  magnetic  field  is  a 
region  in  which  a  magnet  pole  is  pulled  in  a  definite  direction 
(the  direction  of  the  field  in  the  case  of  a  north  pole,  the  oppo- 
site direction  in  the  case  of  a  south  pole).  The  force  H,  in 
dynes,  which  acts  upon  a  unit  magnet  pole  when  it  is  placed  at 
a  given  point  in  a  magnetic  field  is  adopted  as  the  numerical 
measure  of  the  intensity  of  the  field  at  the  point.  This  force-per- 
unit-pole  H  is  hereafter  spoken  of  simply  as  the  intensity  of 
the  field.  The  unit  of  magnetic  field  intensity  (one-dyne-per- 
unit  pole)  is  called  the  gauss. 

Complete  expression  for  the  force  with  which  a  magnetic  field 
acts  on  a  magnet  pole.  —  The  force  with  which  a  magnetic  field 
acts  upon  a  magnet  pole  of  m  units  strength  is  m  times  as  great 
as  the  force  H  with  which  the  field  acts  upon  a  unit  pole  placed 
at  the  same  point.  Therefore 

F=mH  (16) 

in  which  F  is  the  force  in  dynes  which  acts  upon  a  magnet  pole 
of  strength  m  when  it  is  placed  in  a  magnetic  field  of  which  the 
intensity  is  H  gausses. 

Uniform  and  non-uniform  fields.  —  A  magnetic  field  is  said  to 
be  uniform  or  homogeneous  when  it  has  at  every  point  the  same 
direction  and  intensity,  otherwise,  it  is  said  to  be  non-uniform  or 
non-homogeneous.  The  earth's  magnetic  field  is  in  many  places 
sensibly  uniform  throughout  a  room.  The  magnetic  field  sur- 
rounding a  magnet  is  non-uniform.  The  magnetic  field  surround- 
ing an  electric  wire  is  non-uniform. 

36.  Direction  and  intensity  of  the  magnetic  field  surrounding  an 
isolated  magnet  pole.  —  Consider  the  poles  of  two  magnets  of 
which  the  strengths  are  M  and  mt  respectively,  as  shown  in 

*  This  statement  refers  to  the  case  in  which  the  field  is  uniform,  as  will  be  seen 
later. 


68  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Fig.  32.  The  force  F  with  which  the  pole  M  repels  the  pole 
m  is  given  by  equation  (15),  namely,  F=MmjriJ  but  the  force 
which  acts  upon  the  pole  m  is  equal  to  mH  where  H  is  the 


M 


Fig.  32. 

intensity  AT  m  of  the  magnetic  field  which  is  due  to  the  agent 
which  is  exerting  the  force  on  m,  that  is,  where  H  is  the  inten- 
sity of  the  field  at  m  due  to  M.  Therefore  mH=F=Mwi/r2,  or 

*-?  07) 

in  which  H  is  the  intensity  in  gausses  of  the  magnetic  field  at  a 
point  distant  r  centimeters  from  an  isolated  magnet  pole  of 
which  the  strength  is  M.  In  the  neighborhood  of  a  north  pole 
the  magnetic  field  is  directed  away  from  the  pole  and  in  the 
neighborhood  of  a  south  pole  the  magnetic  field  is  directed 
towards  the  pole.  This  is  evident  when  we  consider  that  the 
direction  of  the  field  is  indicated  by  the  direction  in  which  a  com- 
pass needle  would  point,  arrow-head  being  supposed  to  be  on 
the  north-pointing  pole  of  the  needle. 

37.  Representation  of  magnetic  field  intensity  at  a  point  by 
means  of  a  line.  —  The  magnetic  field  intensity  at  a  point,  like 
the  velocity  of  a  fluid  at  a  point,  may  be  represented  by  a  line 
drawn  in  the  direction  of  the  field  at  the  point,  the  length  of  the 
line  being  such  as  to  represent  the  intensity  of  the  field  at  the 
point  to  a  convenient  scale. 

Composition  of  magnetic  fields.  —  Consider  two  agents  which 
acting  singly  produce  magnetic  fields  whose  respective  directions 
and  intensities  at  a  point  /  are  represented  by  the  lines  I  and  2 


THE   MAGNETISM   OF   IRON. 


69 


in  Fig.  33#.  These  two  agents  acting  together  produce  a  mag- 
netic field  at  p  which  is  represented  by  the  line  3  which  is  the 
resultant  of  I  and  2. 

Resolution  of  a  magnetic  field  into  components.  —  Consider  a 
magnetic  field  whose  direction  and  intensity  at  a  point  /,    Fig. 


Fig.  33a- 


Fig.  33b. 


,  is  represented  by  the  line  R.  It  is  often  convenient  to  con- 
sider that  part  of  the  field  which  acts  in  a  given  direction ;  thus, 
Hy  Fig.  33^,  is  called  the  horizontal  component  of  R,  and  V 
is  called  the  vertical  component  of  R. 

38.  Magnetic  flux.  —  Let  a  (expressed  in  square  centimeters) 
be  an  area  at  right  angles  to  the  velocity  of  a  moving  liquid  and 
let  v  (expressed  in  centimeters  per  second)  be  the  velocity  of 
the  liquid.  Then  av  is  the  flux  of  the  liquid  across  the  area  in 
cubic  centimeters  per  second.  Thus,  if  v  is  the  velocity  of 
liquid  in  a  pipe  and  a '  is  the  sectional  area  of  the  pipe,  then  av 
is  the  number  of  cubic  centimeters  of  liquid  discharged  per 
second  by  the  pipe. 

Similarly,  the  product  of  the  intensity  H  of  a  magnetic  field 
and  an  area  a  at  right  angles  to  H  is  called  the  magnetic  flux 
across  the  area  ;  that  is, 

3>  =  aH  (18) 

in  which  <3>  is  the  magnetic  flux  across  an  area  of  a  square  centi- 
meters which  is  at  right  angles  to  a  magnetic  field  of  which  the 
intensity  is  H  gausses. 

Representation  of  the  magnetic  flux  across  an  area  by  the  number 
of  lines  of  force  which  pass  through  the  area.  Imagine  a  surface 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


drawn  across  a  magnetic  field,  the  surface  being  at  each  point  at 
right  angles  to  the  field.  Of  course,  this  chosen  surface  will  be 
curved  if  the  lines  of  force  are  not  parallel  straight  lines,  which, 
in  general,  they  are  not.'  Imagme  lines  of  force  drawn  through 
the  field  so  that  the  number  of  lines  which  pass  through  each 
square  centimeter  of  this  surface  is  equal  to  the  intensity  of  the 
magnetic  field  at  that  part  of  the  surface.  Then  the  magnetic  flux 
passing  through  any  area  anywhere  in  the  field  will  be  equal  to 
the  number  of  these  lines  that  cross  that  area.  The  unit  of  flux 
(that  is,  the  flux  across  a  square  centimeter  at  right  angles  to  a 
field  of  which  the  intensity  is  one  gauss)  is  therefore  called  the 
line  of  force  or  simply  the  line,  and  a  magnetic  flux  is  usually 
specified  as  so  many  lines.  The  name  maxwell  has,  however,  been 
internationally  adopted  as  the  name  for  the  unit  of  magnetic  flux. 

39.  Total  magnetic  flux  emanating  from  a  magnet  pole  of  strength 
M.  Proposition.  —  The  number  of  lines  of  force  (the  number  of 
maxwells  of  flux)  which  emanate  from  a  magnet  pole  of  strength 
Mis 

(19) 


Proof.  —  Imagine  a  spherical  surface  of  radius  r  drawn  with 

the  pole  M  at  its  center,  as  rep- 
resented by  the  dotted  line  in  Fig. 
34.  The  area  of  this  spherical 
surface  is  ^rrr2  (neglecting  the 
small  portion  of  the  sphere  which 
falls  inside  of  the  material  of  the 
slim  magnet  at  the  point  fr)  ;  the 
magnetic  field  at  the  spherical 
surface  due  to  the  pole  M  is 
everywhere  at  right  angles  to 
the  surface,  and  its  intensity  is  ev- 
erywhere equal  to  M/r2,  according  to  equation  (17).  Therefore, 
according  to  equation  (18),  the  magnetic  flux  <I>  across  the  spher- 
ical surface  is  equal  to  4-Trr2  times  J///2,  which  is  equal  to  477  J/. 


Fig.  34. 


THE   MAGNETISM   OF   IRON. 


General  relation  between  pole  strength  and  flux.  —  A  magnet 
pole  may  be  defined  as  a  place  where  magnetic  lines  of  force  pass 
from  iron  into  air  (north  pole)  or  from  air  into  iron  (south  pole). 
A  piece  of  iron  may  be  magnetized 
so  that  the  magnetic  flux  does  not 
pass  out  of  the  iron.  In  such  a 
case,  there  are  no  magnetic  poles. 
Thus,  the  iron  ring  shown  in  Fig. 
35  has  no  magnetic  poles  when  it 
is  magnetized  by  a  current  flowing 
through  the  winding  of  wire. 

The  relation  between  pole  strength 
and  magnetic  flux  which  is  given  in 
equation  (19)  is  entirely  general ; 
47r;;z  lines  of  force  emanate  from  any  north  pole  of  which  the 
strength  is  m,  whatever  the  shape  and  size  of  the  pole  may  be  ; 
and  47r;/z  lines  of  force  converge  upon  any  south  pole  of  which 
the  strength  is  m. 

40.  Magnetic  field  in  the  neighborhood  of  a  long  slim  pole.  — 

Consider  a  long  slim  magnet  having  one  of  its  poles  spread  uni- 
formly over  /  centimeters  of  its  end,  as  indicated  by  the  shaded 
area  in  Fig.  36. 

The  lines  of  force  emanate  from  this  uniformly  distributed  pole 
in  planes  at  right  angles  to  the  axis  of  the  rod  as  indicated  by 

_ I  centimeters 


Fig.  35. 


steel 


firtiit 


rod 


$^$$$9%$%^^ 


aide  view 


end  view 


Fig.  36. 


the  fine  lines  in  Fig.  36,  and  the  intensity   H  of  this  field  at  a 
point  distant   r   centimeters  from  the  axis  of  the  rod  is 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


21   —  . 

(20) 


in  which  mjl  is  the  pole  strength  per  unit  length  of  the  shaded 
area  in  Fig.  36,  and  H  is  the  field  intensity  in  gausses  at  a  point 
r  centimeters  from  the  axis  of  the  rod. 

Proof.  —  Imagine  a  cylindrical  surface  of  radius  r  to  be  drawn 
with  its  axis  coincident  with  the  axis  of  the  rod,  as  shown  by  the 
dotted  lines  in  Fig.  36.  The  area  of  this  cylindrical  surface  is  equal 
to  27rr/,  and,  inasmuch  as  the  magnetic  field  at  this  cylindrical  sur- 
face is  everywhere  at  right  angles  to  it  and  everywhere  the  same 
value,  the  magnetic  flux  through  this  cylindrical  surface  is  equal  to 
2irrl  x  H  according  to  equation  ( 1 8).  This  is  the  total  magnetic 
flux  emanating  from  the  pole,  and  it  must  be  equal  to 
according  to  equation  (19),  so  that  we  have  2irrlH '  — 
whence  H  =  2mjrl.  In  this  discussion  the  non-uniformity  of 
the  magnetic  field  near  the  ends  of  the  long  slim  pole  is  ignored ; 
in  fact,  the  effect  of  this  non-uniformity  is  negligible  if  r  is  small 
in  comparison  with  the  length  /  of  the  slim  pole. 

The  above  formula  expressing  the  field  intensity  at  a  distance 
from  a  long  slim  pole  applies  also  to  the  case  of  the  pole  which  is 
distributed  along  the  edge  of  a  steel  ribbon  which  is  magnetized 
crosswise  as  shown  in  Fig.  37.  In  this  case,  however,  the 


jv AT          j\r         jv          #  jv 


S  S  5  S 

side  uiew  end  view 

Fig.  37. 

actual  magnetic  field  at  any  given  point  /   is  the  resultant  of  the 
fields  due  to  the  poles  along  both  edges  of  the  strip. 


THE   MAGNETISM    OF   IRON.  73 

41.  Behavior  of  a  magnet  in  a  uniform  magnetic  field. — A  bar 

of  steel  weighs  the  same  before  and  after  being  magnetized 
(earth's  field  being  uniform),  and  the  fiber  by  which  a  magnet  is 
suspended  hangs  vertically  (earth's  field  being  uniform).  Any 
force  tending  to  produce  translatory  motion  of  a  magnet  would 
cause  it  to  weigh  more  or  less  after  magnetization  than  before, 
or  would  tend  to  cause  a  suspending  fiber  to  be  out  of  plumb. 
Therefore  the  forces  with  which  the  uniform  magnetic  field  of  the 
earth  acts  upon  a  magnet  do  not  tend  to  produce  translatory 
motion,  the  force  which  acts  on  the  north  pole  of  the  magnet  is 
equal  in  value  and  opposite  in  direction  to  the  force  which  acts 
upon  the  south  pole  of  the  magnet,  as  indicated  in  Fig.  38,  and 
therefore  the  poles  of  the  magnet  are  equal  in  strength  and 
opposite  in  sign. 

Consider  a  magnet  of  length    /  placed  in  a  uniform  magnetic 
field  of  intensity   Ht    the  angle  between  the  axis  of  the  magnet 
and    the    direction    of  the 
field  being  6,  as  shown  in 
Fig.  38.     The  poles  of  the 
magnet  are  acted  upon  by  /&    \         lines  of  force 

,     ^    ,  rr    ^  j$r~~*T~--     Of  fold  ff 

the  forces  -f  mH  and  — 
mH,  respectively,  the  mo- 
ment of  each  of  these  forces 

Fig.  38. 

about    the    center    of    the 

magnet  is  equal  to  mH  x  //  2  x  sin  6,  and  both  of  these  moments 
tend  to  turn  the  magnet  in  the  same  direction.  Therefore  the 
total  torque  T  tending  to  turn  the  magnet  into  the  direction  of 

the  field  is 

T=—mlHsmO  (21) 

The  negative  sign  is  chosen  simply  for  the  reason  that  the  torque 
tends  to  reduce  6  which  may  be  considered  as  a  positive  angle. 
This  equation  expresses  the  torque  in  dyne-centimeters.  When 
the  angle  6  is  equal  to  zero  or  180°,  the  torque  T  is  zero  and 
the  forces  -f  mH,  —  mH  have  no  tendency  to  turn  the  magnet, 


74          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

that  is  to  say,  the  magnet  is  in  equilibrium.  This  equilibrium  is 
stable  when  the  north  pole  of  the  magnet  points  in  the  direction 
of  the  magnetic  field  (9  equal  to  zero),  and  it  is  unstable  when 
the  south  pole  of  the  magnet  points  in  the  direction  of  the  mag- 
netic field  (6  equal  to  180°). 

If  the  angle    6    is  never  large  in  value  then    6    (in  radians) 
may  be  written  for    sin  0    in  equation  (21)  givihg 

T=-mlH-e  (22) 

This  equation  shows  *  that  a  suspended  magnet  when  started  will 
perform  harmonic  vibrations  about  its  axis  of  suspension  in  such 
a  manner  that 

4~^mlH  (23) 

in  which  K  is  the  moment  of  inertia  of  the  magnet  about  the 
axis  of  suspension,  and  t  is  the  period  of  one  complete  vibra- 
tion. This  equation  is  not  even  approximately  true  if  6  reaches 
large  values,  that  is,  if  the  amplitude  of  the  oscillations  of  the 
magnet  is  large. 

42.  Gauss's  method  for  measuring  the  horizontal  component  of 
the  earth's  magnetic  field.  —  A  method  was  devised  by  Gauss  in 
1850  for  determining  the  value  of  the  horizontal  component  of  the 
earth's  magnetic  field.     The  details  of  this  method  are  described 
in  Chapter  X. 

43.  Behavior  of  a  magnet  in  a  non-uniform  magnetic  field.  — 
The  forces  which  act  upon  the  poles  of  a  magnet  in  a  non-uni- 
form magnetic  field  tend  in  general  to  turn  the  magnet  and  also 
to  impart  to  it  a  motion  of  translation,  because  the  force  which 
acts  on  the  north  pole  of  the  magnet  is  in  general  not  opposite 
in  direction  and  not  equal  in  value  to  the  force  which  acts  on  the 
south  pole  of  the  magnet ;  that  is,  the  field  at  the  north  pole  of 
the  magnet  is  in  general  different  in  intensity  and  in   direction 
from  the  field  at  the  south  pole  of  the  magnet.     This  is  shown 

*  See  discussion  of  harmonic  motion  in  any  good  treatise  on  elementary  mechanics. 


THE   MAGNETISM   OF   IRON. 


75 


in  Fig.  39  where  a  small  magnet  is  placed  in  the  non-uniform 
field  near  the  pole  of  a  large  magnet.  The  forces  F  and  F' 
are  different  in  value  and  not  opposite  in  direction. 

The  attraction  of  a 
particle  of  iron  by  a 
magnet  depends  in  the 
first  place  upon  the  mag- 
netization of  the  particle 
of  iron  and  in  the  sec- 
ond place  upon  the  non- 
uniformity  of  the  mag- 
netic field  in  which  the 
magnetized  particle  finds 

itself,  that  is  to  say,  the  p.g  39> 

particle  of  iron  becomes 

a  magnet  and  its  two  poles  are  acted  upon  by  unequal  forces 
on  account  of  the  non -uniformity  of  the  field. 


Fig.  40. 


Fig.  41. 


The  magnetic  field  near  a  flat-ended  magnet  pole  is  approxi- 
mately uniform  (lines  of  force  parallel  straight  lines)  as  shown  in 
Fig.  40 ;  near  the  sharp  corners  of  the  pole,  however,  the  field  is 
distinctlynon-uniform  (lines  of  force  diverge  strongly).  Therefore 
particles  of  iron  are  not  perceptibly  attracted  by  the  flat-face  of  the 
pole  whereas  the  sharp  corners  of  the  pole  attract  particles  of  iron 


76  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

very  strongly.  This  is  shown  very  strikingly  by  passing  the  flat 
end  of  a  magnet  pole  over  a  table  on  which  a  very  few  iron  filings 
have  been  placed,  the  filings  are  all  caught  by  the  corners  of  the 
pole. 

The  lines  of  force  in  the  neighborhood  of  a  sharp-pointed  mag- 
net pole  diverge  very  greatly  indeed  as  shown  in  Fig.  41,  that  is 
to  say,  the  magnetic  field  in  the  neighborhood  bf  the  point  is  non- 
uniform  to  a  high  degree,  and 

£        MIXED  .  .  . 

$  MATERIAL  such  3.  magnet  pole  has  a 
strong  attraction  for  small 
particles  of  magnetic  material. 


MAGNET  POLE          \>   I  A  pointed  magnet  pole  is  an 

essential  feature  of  the  mag- 


netic  ore  separator,  the  action 


MAGNETIC     .--'  i'  NON-MAGNETIC 

MATERIAL  £  |         MATERIAL 

The  crushed  ore  falls  in  a  thin 

F1  stream  before  a  pointed,  or 

wedge-shaped,  magnet  pole. 

The  particles  of  magnetic  material  are  attracted  by  the  pointed 
pole  and  thus  deflected,  while  the  non -magnetic  material  falls 
straight  downwards. 

Surgeons  sometimes  make  use  oY  a  pointed  magnet  for  remov- 
ing particles  of  iron  or  steel  from  the  eye. 

44.  Tension  and  energy  of  the  magnetic  field.  —  Consider  the 
opposite  poles  of  two  magnets  as  shown  in  Fig.  43.  Their  force 
of  attraction  is  due  to  the  tension  of  the  magnetic  field,  the  ten- 
sion of  the  lines  of  force  as  it  is  sometimes  called.  The  lines  of 
force  of  the  magnetic  field  also  push  each  other  apart  side  wise. 
This  sidewise  push  of  the  lines  of  force  on  each  other  is  evident  if 
we  consider  that  the  lines  of  force  in  Fig.  43  are  curved  so  that 
they  must  exert  a  side  force  if  they  are  under  tension. 

When  the  two  magnet  poles  in  Fig.  43  are  allowed  to  move 
nearer  together,  their  force  of  attraction  does  mechanical  work, 
and  the  mechanical  work  thus  obtained  comes  from  the  magnetic 
field  ;  that  is  to  say,  a  magnetic  field  represents  a  store  of  energy, 


THE   MAGNETISM    OF    IRON. 


77 


and  when  a  magnetic  field  is  reduced  *  in  extent  (volume)  or  in 
intensity,  a  portion  of  its  energy  is  transformed. 

A  simple  discussion  of  the  tension  and  energy  of  the  magnetic 
field  cannot  be  based  on  an  arrangement  like  Fig.  43  because  of 
the  non-uniformity  of  the  field.  Consider  one  end  of  a  very 
broad  flat  strip  of  magnetized  steel,  as  shown  in  Fig.  44,  and  let 


Fig.  43. 

us  assume  that  the  total  pole  strength  m  is  spread  uniformly  over 
the  end  of  the  strip  f  as  indicated  by  the  shading  in  the  figure. 

*  When  the  two  poles  in  Fig.  43  move  nearer  together  the  intensity  of  the  inter- 
vening field  is  increased  in  some  parts  and  decreased  in  other  parts. 

•(•The  fundamental  relations  involved  in  the  study  of  electricity  and  magnetism  may 
be  established  in  a  comparatively  simple  way  by  assuming  simply  geometrical  forms  and 
distributions.  Thus,  the  formula  expressing  the  magnetic  field  intensity  in  the  neigh- 
borhood of  a  magnet  pole  is  extremely  complicated  unless  the  pole  be  assumed  to  be 
concentrated  at  a  point,  or  to  be  spread  uniformly  over  a  certain  length  of  a  rod,  or  to 
be  spread  uniformly  over  a  certain  plane  area.  The  formula  expressing  the  intensity 
of  a  magnetic  field  in  the  neighborhood  of  a  wire  carrying  an  electric  current  is 
extremely  complicated  unless  the  wire  be  simple  in  shape.  Thus,  the  formula  express- 
ing the  intensity  of  a  magnetic  field  in  the  neighborhood  of  a  long  straight  wire  is  very 


78  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Magnetic  lines  of  force  emanate  from  both  faces  of  the  polar  area 
s  as  shown  in  the  edge  view  in  Fig.  44,  and  the  magnetic  field  on 
each  side  of  the  flat  pole  is  a  uniform  field  (except,  of  course,  near 


rty 


polar  area 

s_  square 

centimeters 


fiat  view 


Fig.  44. 


the  edges,  but  the  polar  area  is  assumed  to  be  so  large  that  the 
edge  complications  may  be  ignored).  Let  Hl  be  the  intensity 
of  this  field.  Then  H^s*  is  the  magnetic  flux  passing  out  from  the 
polar  area  on  each  side,  2H^s  is  the  total  flux  emanating  from 
the  pole,  and  this  must  be  equal  to  47r;;/  according  to  Art.  39, 
so  that  we  find  : 

(0 


Consider  two  similar  flat  magnet  poles  AB  and  A' Bf  placed 
side  by  side  as  shown  in  Fig.  45,  one  being  a  north  pole  and  the 
other  a  south  pole,  as  indicated  in  the  figure.  Consider  the  mag- 
simple.  These  simple  modes  of  distribution  of  magnet  pole,  and  long  straight  wires 
carrying  electric  currents  are  never  met  with  as  actual  facts,  but  they  are  possible  and 
therefore  legitimate  as  starting  points  for  the  development  of  simple  mathematical 
theory. 

*  This  expression  ignores  the  non-uniformity  of  the  field  near  the  edges  of  the  flat 
pole. 


THE   MAGNETISM    OF    IRON. 


79 


netic  field  which  is  due  to  AB,  its  intensity  is  equal  to  2irm  js 
throughout  the  whole  region  occupied  by  the  pole  A' B' ,  ac- 
cording to  equation  (i),  and  therefore  the  force  which  is  exerted 
upon  A1  B'  is  equal  to  the  product  of  the  strength  of  A' B' 
and  the  intensity  of  the  field  due  to  AB,  whence  we  find : 


in  which    F  is  the  force  in  dynes  with  which  the  two  poles  in 
Fig.  45  attract  each  other.     It  is  noteworthy  that  this  force  is 
independent  of  the  distance  d,   provided 
the  distance  d  is  small  in  comparison 
with  the  length  and  breadth  of  the  polar 
areas  AB  and  A' B' . 

To  find  the  intensity  of  the  field  in  the 
region  between  the  flat  poles  in  Fig.  4.5. — 
The  north  pole  AB,  Fig.  45,  tends  to 
produce  in  the  region  RR  a  uniform  mag- 
netic field  directed  towards  the  left,  of 
which  the  intensity  is  27rm/s,  whereas 
the  south  pole  A' B'  tends  to  produce  in 
the  region  RR  a  uniform  magnetic  field 
directed  towards  the  right t  of  which 
the  intensity  is  27rm/s,  and  the  net  re- 
sult is  that  the  magnetic  field  intensity 
in  the  region  RR  is  zero,  or,  in  other 
words,  no  lines  of  force  traverse  the  re- 
gion RR.  In  a  similar  manner  it  can  be  shown  that  no  lines  of 
force  traverse  the  region  R'Rf.  In  the  region  between  AB  and 
A' B'  each  magnet  pole  tends  to  produce  a  magnetic  field  towards 
the  right  of  which  the  intensity  is  27rm/s,  so  that  the  actual  in- 
tensity H  of  the  field  between  AB  and  A1  B1  is 


1 

8 

x> 
5 

> 

>r 

< 

s 

w 

1 

§ 

1 

kh 

R" 
ST-po/e 
R' 

>J  - 

»— 

•    .  > 

Bf 



"*• 

C  D 

edge  view  of  two  fiat  poles 

Fig.  45. 


(25) 


So 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


The  arrangement  in  Fig.  4.5  is  equivalent  to  the  arrangement 
shown  in  Fig.  4.6.  —  In  the  arrangement  shown  in  Fig.  45  the 
magnetic  flux  which  crosses  from  AB  to  A' B'  comes  up 
through  the  steel  at  C  and  goes  down  through  the  steel  at  D. 
Figure  46  shows  the  flat  ends  of  two  massive  steel  or  iron  bars 
which  are  magnetized  so  that  the  face  of  one  bar  is  a  north  pole 


steel 


steel 


N 


steeL 


steel 


Fig.  46. 

and  the  face  of  the  other  bar  is  a  south  pole  as  indicated.  In  this 
case,  the  magnetic  flux  comes  up  to  the  polar  areas  through  the 
steel  at  C  and  Dy  Fig.  46.  The  two  magnet  poles  in  Fig.  46 
act  on  each  other  in  the  same  way  as  the  two  magnet  poles  in 
Fig.  45,  and  equations  (24)  and  (25)  apply  to  both  figures. 

Tension  of  the  lines  of  force. — The  force  attraction  of  the  two 
poles  in  Figs.  45  and  46  is  due  to  the  tension  of  the  lines  of  force. 
It  is  desirable  to  express  this  tension  in  terms  of  the  field  inten- 
sity, and  for  this  purpose  the  force  of  attraction  of  the  two  poles 
must  be  expressed  in  terms  of  the  field  intensity  between  them, 
instead  of  being  expressed  in  terms  of  the  strengths  of  the  two 
poles,  as  in  equation  (24).  The  strength  of  each  pole  may  be 
expressed  in  terms  of  the  intensity  of  the  field  between  the  poles 
by  solving  equation  (25)  for  m.  This  value  of  m  may  then  be 
substituted  in  equation  (24),  giving 

^_2  .„, 

~~    STT 


THE   MAGNETISM   OF   IRON.  8  1 

Dividing  both  members  of  this  equation  by  the  sectional  area  of 
the  region  between  poles  in  Figs.  45  and  46,  we  get  the  force  per 
unit  area  which  is  transmitted  across  the  region,  or  in  other  words, 
the  tension  of  the  magnetic  field.  Therefore 

Tension  of  a  magnetic  field  in  ]         H 

dynes  per  square  centimeter  )         g  77-  \      f 

in  which    H  is  the  intensity  of  the  field  in  gausses. 

Energy  of  the  magnetic  field.  —  If  the  magnet  poles  in  Fig.  45 
or  46  are  allowed  to  move  together,  their  force  of  attraction  will 
do  an  amount  of  work,  W  =  Fd,  where  d  is  the  initial  distance 
apart  of  the  two  pole  faces,  and  the  mechanical  work  thus  gained 
comes  from  the  magnetic  field  that  existed  in  the  air  space. 
Therefore,  using  the  value  of  F  from  equation  (ii)  we  have 


but   sd  is  the  volume  of  the  region  between  the  poles,  so  that 

Energy  of  a  magnetic  field  in  1         H 


ergs  per   cubic  centimeter  j 


\   7) 


45.  The  magnetization  of  iron.*  —  When  a  piece  of  iron  or  other 
magnetic  substance,  such  as  cobalt  or  nickel,  is  placed  in  a  mag- 
netic field,  it  becomes  a  magnet.  For  example,  a  neutral  or 
unmagnetized  bar  of  iron  or  steel  when  held  in  the  direction  of 
the  earth's  magnetic  field  shows  north  polarity  at  one  end  and 
south  polarity  at  the  other  end  (the  polarity  of  the  bar  may  be 
indicated  by  a  compass  needle).  If  the  bar  is  turned  end  for  end 
its  magnetism  is  reversed.  A  sharp  blow  with  a  hammer  renders 
the  bar  more  susceptible  to  the  influence  of  the  weak  magnetic 
field  of  the  earth.  This  action  of  a  magnetic  field  upon  iron  is 
called  magnetisation. 

When  a  piece  of  iron  is  placed  in  a  magnetic  field  the  trend  of 

*  For  a  full  discussion  of  the  theory  of  the  magnetization  of  iron  the  student  is  re- 
ferred to  Franklin  and  Esty's  Elements  of  Electrical  Engineering,  Vol.  I,  Appendix 
A  ;  to  J.  A.  Ewing's  Magnetic  Induction  in  Iron  and  Other  Metals,  London,  1900  ; 
and  to  H.  DuBois'  Magnetic  Circuit  in  Theory  and  Practice,  translated  by  Atkinson, 
New  York,  1896. 

7 


82 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


the  lines  of  force  in  the  field  is  greatly  altered  ;  in  fact,  the  field 
becomes  the  resultant  of  two  fields,  namely,  the  original  field  and 


:\ 


Fig.  47. 


the  field  due  to  the  piece  of  iron  which  has  become  a  magnet. 
Thus,  Fig.  47  shows  the  effect  of  a  small  piece  of  iron  upon  the 


Fig.  48. 


magnetic  field  between  two  flat-ended  magnet  poles.     In  the  ab- 
sence of  the  iron  the  field  is  as  shown  in  Fig.  48.     The  effect  of 


THE   MAGNETISM   OF   IRON.  83 

a  piece  01  iron  in  a  magnetic  field  is  always  such  as  to  suggest 
that  "iron  is  a  better  carrier  of  lines  of  force  than  air."  The 
lines  of  force  tend  to  converge  into  the  iron  and  pass  through  it. 
Magnetic  screening.  —  A  shell  of  soft  iron  forms  a  very  effec- 
tive screen  which  protects  the  region  inside  of  the  shell  from  the 
action  of  outside  magnetic  influences.  The  lines  of  force,  which 
would  pass  through  the  region  occupied  by  the  shell  if  the  shell 
were  not  present,  pass  into  the  iron  and  tend  to  flow  around 
through  the  shell  and  pass  out  on  the  other  side  without  crossing 
the  region  inside  of  the  shell.  This  screening  effect  has  been 
used  for  the  protection  of  watches  against  magnetic  disturbances 
by  providing  the  watch  with  a  thick  case  of  soft  iron. 

Note.  —  The  region  surrounding  a  magnet  is  a  magnetic  field,  it  magnetizes  any 
piece  of  iron  in  the  neighborhood,  and  the  piece  of  iron  is  then  attracted  by  the 
magnet. 

46.  Residual  magnetism.  Permanent  magnets.  —  An  iron  rod 
retains  much  of  its  magnetism  when  it  is  removed  from  a  mag- 
netic field  in  which  it  has  been  magnetized  ;  or  in  case  of  an  elec- 
tromagnet, when  the  magnetizing  current  is  reduced  to  zero. 
Long  slim  bars  retain  a  greater  portion  of  their  magnetism  than 
short  thick  bars,  because  of  the  fact  that  in  short  bars  the  poles 
of  the  magnet  are  closer  together  and  produce  of  themselves  a 
strong  demagnetizing  field  along  the  bar.  The  magnetism  which 
is  thus  left  in  a  bar  of  iron  or  in  an  electromagnet  is  called  resid- 
ual magnetism.  Long  slim  bars  of  annealed  wrought  iron  may 
retain  in  this  way  as  much  as  90  per  cent,  of  their  magnetism, 
but  a  very  weak  demagnetizing  field  or  a  very  slight  mechanical 
shock  is  sufficient  to  cause  such  a  bar  to  lose  its  residual  mag- 
netism almost  completely.  Cast  iron,  hard  drawn  iron  wire  and 
mild  steel  retain  a  smaller  portion  of  their  magnetism  but  with 
greater  persistence,  and  hardened  steel  bars  retain  a  portion  of 
their  magnetism  very  persistently  even  when  roughly  handled. 
Magnetized  bars  of  hardened  steel  are  called  permanent  magnets. 

Aging  of  permanent  magnets.  —  A  freshly  magnetized  bar  of 
hardened  steel  loses  a  portion  of  its  residual  magnetism  rapidly 


84          ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

when  it  is  subjected  to  mechanical  shocks  or  to  changes  of  tem- 
perature. After  the  residual  magnetism  has  been  reduced  in  this 
way,  a  remainder  is  left  which  decreases  but  little  with  repeated 
mechanical  shocks  and  changes  of  temperature,  and  the  magnet 
is  said  to  be  aged.  Permanent  magnets  for  use  in  electrical 
measuring  instruments  are  always  subjected  to  an  aging  process 
which  consists,  usually,  in  placing  the  magnet  repeatedly  in  hot 
and  then  in  cold  water,  and  in  subjecting  it  to  a  series  of  slight 
mechanical  shocks. 

Demagnetization.  —  When  iron  is  heated  to  bright  redness  it 
loses  its  magnetic  properties.  Thus,  red  hot  iron  is  not  attracted 
by  a  magnet  When  a  magnetized  bar  of  steel  is  heated  to 
bright  redness  its  magnetization  disappears  and  the  bar,  upon 
cooling,  is  found  to  be  completely  demagnetized. 

Any  piece  of  iron  or  steel  may  be  completely  demagnetized  by 
the  following  operation  :  Place  the  piece  of  iron  or  steel  in  a  coil 
of  wire  through  which  a  strong  electric  current  is  flowing.  Re- 
verse the  current  repeatedly  and  at  the  same  time  slowly  reduce 
its  value  to  zero.  This  operation  is  called  demagnetization  by 
reversals.  A  watch  which  has  been  disturbed  by  a  strong  mag- 
netic field  is  usually  demagnetized  by  this  process. 

47.  Intensity  of  magnetization.     Magnetic  saturation.  —  Let  m 

be  the  strength  of  the  magnetic  pole  at  the  end  of  an  iron  rod  of 
which  the  sectional  area  is  s  square  centimeters.  The  ratio  mjs 
is  called  the  intensity  of  magnetization  of  the  rod.  When  an 
iron  rod  is  subjected  to  a  stronger  and  stronger  magnetizing  field, 
its  magnetization  becomes  more  and  more  intense  and  approaches 
a  definite  limiting  value  beyond  which  it  cannot  be  magnetized 
however  strong  the  magnetizing  field  may  be.  The  iron  rod  is 
said  to  approach  magnetic  saturation  as  it  approaches  this  limit- 
ing intensity  of  magnetization.  The  limiting  value  of  mfs  is 
about  1,730  units  of  pole  per  square  centimeter  of  section  for 
wrought  iron,  about  1,600  for  mild  steel,  about  1,310  for  cobalt, 
and  about  540  for  nickel.  Permanent  magnets  of  hardened  steel 


THE   MAGNETISM   OF   IRON.  85 

have  at  the  utmost  about  800  units  pole  per  square  centimeter 
of  section. 

48.  The  molecular  theory  of  the  magnetization  of  iron.  —  When 
a  magnet  is  broken  in  pieces,  each  piece  is  found  to  be  a  complete 
magnet  having  a  north  pole  and  a  south  pole.  This  fact  sug- 
gests the  possibility  that  each  molecule  of  iron  may  be  a  magnet. 
Indeed,  the  hypothesis  that  each  molecule  of  iron,  or  any  sub- 
stance capable  of  being  magnetized,  is  a  permanent  magnet  leads 
to  a  very  useful  conception  of  what  takes  place  in  a  bar  of  iron 
when  it  is  magnetized. 

Explanation  of  magnetization.  —  In  unmagnetized  iron  or  steel 
the  molecular  magnets  are  thought  of  as  pointing  at  random  in 
all  directions,  thus  neutralizing  each  other.  When  the  iron  or 
steel  is  placed  in  an  intense  magnetic  field,  the  molecular  mag- 
nets are  turned  with  their  axes  parallel  to  the  field,  their  north 
poles  all  in  one  direction,  and  the  iron  or  steel  is  completely 
magnetized  or  saturated.  If  the  magnetizing  field  is  weak  the 
molecular  magnets  are  only  partially  turned  and  the  iron  is  only 
partially  magnetized. 

Explanation  of  retention  of  magnetization.  —  A  bar  of  iron 
which  is  strongly  magnetized,  does  not  return  to  its  original 
state  when  the  magnetizing  field  ceases  to  act.  This  is  analo- 
gous to  the  production  of  a  permanent  set  when  an  imper- 
fectly elastic  substance  is  greatly  distorted.  This  persistence  of 
a  portion  of  the  magnetization  in  a  strongly  magnetized  bar  may 
be  ascribed  to  a  friction-like  opposition  to  the  rotation  of  the 
molecular  magnets.  In  annealed  iron  this  friction  is  small,  in 
hard  drawn  iron  wire  it  is  greater,  and  in  hardened  steel  it  is  very 
great.  Mechanical  vibration  and  rise  of  temperature  both  act  as 
if  to  decrease  this  frictional  resistance,  thus  enabling  a  given 
magnetizing  field  to  produce  more  intense  magnetization  and 
causing  the  residual  magnetism  to  disappear. 

Behavior  of  iron  and  steel  when  subjected  to  slight  changes  of 
magnetization.  —  When  a  bar  of  iron  or  steel  is  placed  in  a  weak 


86  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

magnetizing  field,  it  returns  almost  completely  to  its  initial  con- 
dition when  the  weak  field  ceases  to  act.  A  bar  of  iron 
01  steel,  which  is  placed  in  a  strong  magnetizing  field,  re- 
turns almost  completely  to  ifc  initial  condition  when  the  field 
is  slightly  increased  and  then  decreased  again.  That  is,  a  bar 
of  iron  or  steel  exhibits  a  kind  of  magnetic  elasticity.  This 
action  is  especially  prominent  in  hardened  steel.  Thus,  a  small 
magnet  ns,  Fig.  49,  is  repelled  by  the  strong  north  pole  N 
of  another  magnet.  But  when  the  small  magnet  is  brought  very 
near  to  N,  as  shown  in  Fig.  50,  its  magnetism  is  reversed  and 


Fig.  49.  Fig.  50. 

it  is  attracted  by  N,  and  then,  if  the  reversal  of  magnetization 
of  ns  has  not  been  carried  too  far,  it  will  be  found  to  be  again 
repelled  by  N  when  it  is  removed  to  the  position  shown  in 
Fig.  49.  This  shows  that  the  magnetism  of  ns  after  having 
been  actually  reversed  by  the  field  near  N  returns  approxi- 
mately to  its  initial  value  when  this  reversing  field  ceases  to  act 
This  is  analogous  to  the  following  :  A  flat  steel  spring  is  fixed  in 
a  vise  and  bent  sufficiently  to  give  it  a  permanent  set  to  the  left, 
a  force  is  then  exerted  on  the  rod  bending  it  to  a  slight  extent  to 
the  right  and  when  this  force  ceases  to  act,  the  rod  again  takes 
on  its  "permanent"  bend  to  the  left. 

Swing's  theory.  —  The  apparent  frictional  and  elastic  opposition 
to  the  turning  of  molecular  magnets  may  both  be  ascribed  to  the 
mutual  action  of  these  molecules  as  magnets.  This  was  first 
pointed  out  by  Ewing*  who  constructed  a  model  consisting  of  a 
large  number  of  small  magnets  supported  on  jewels  and  pivots 
and  arranged  on  a  board.  When  this  system  of  magnets  is  sub- 
jected to  the  action  of  a  weak  magnetic  field,  each  magnet  is 
slightly  turned,  and  every  magnet  returns  to  its  initial  position 

*See  Philosophical  Magazine,  series  5,  Vol.  30,  page  205. 


THE    MAGNETISM    OF   IRON.  87 

when  the  field  ceases  to  act.  If  the  field  is  increased  in  intensity 
more  and  more  the  magnets  turn  more  and  more  until  the  con- 
figuration of  the  system  becomes  unstable,  when  the  magnets 
suddenly  fall,  as  it  were,  into  a  new  configuration.*  If  now  the 
field  is  slowly  reduced  in  intensity  the  magnets  tend  to  persist  in 
their  new  configuration. 

49.  Paramagnetic  substances  and  diamagnetic  substances. — Cobalt  and  nickel 
are  similar  to  iron  in  their  magnetic  properties  except  that  the  limit  or  saturation  value 
of  their  intensity  of  magnetization  is  not  so  great.  Many  other  substances,  such  as 
manganese,  chromium,  platinum,  and  oxygen,  show  similar  properties  but  to  a  lesser 
degree.  Such  substances  are  said  to  be  paramagnetic,  or  simply  magnetic.  On  the 
other  hand,  substances  such  as  bismuth,  antimony,  zinc  and  lead,  when  they  are  near 
a  magnet,  are  magnetized  in  such  a  way  as  to  be  repelled  f  by  the  magnet.  Such  sub- 
stances are  said  to  be  diamagnetic. 

Paramagnetic  substances  are  better  carriers  of  lines  of  force  than  air  and  diamag- 
netic substances  are  poorer  carriers  of  magnetic  lines  of  force  than  air,  that  is  to  say, 
when  a  paramagnetic  substance  is  placed  in  a  magnetic  field  the  lines  of  force  converge 
towards  it  and  pass  through  it,  and  when  a  diamagnetic  substance  is  placed  in  a  mag- 
netic field  the  lines  of  force  tend  to  spread  out  and  go  round  it. 

A  paramagnetic  substance  when  placed  in  a  non-uniform  magnetic  field  is  drawn 
towards  the  region  where  the  field  is  most  intense,  whereas  a  diamagnetic  substance 
when  placed  in  a  non-uniform  magnetic  field  is  drawn  towards  the  region  where  the 
field  is  least  intense.  This  behavior  of  a  diamagnetic  substance  in  a  non-uniform 
field  may  be  shown  by  suspending  a  very  small  bar  of  bismuth  between  the  pointed 
poles  of  a  strong  electromagnet.  If  the  suspending  fiber  is  sufficiently  flexible  the  bar 
of  bismuth  sets  itself  at  right  angles  to  the  lines  joining  the  two  pointed  poles.  J 

*  A  group  of  magnets  mounted  on  pivots  may  be  in  equilibrium  in  a  great  variety 
of  configurations. 

|  See  note  in  Art.  45,  page  83. 

J  A  bar  of  bismuth  tends  to  place  itself  parallel  to  the  lines  of  force  in  the  uniform 
magnetic  field  the  same  as  a  bar  of  iron.  This  apparently  similar  property  of  bismuth 
and  iron  may  be  explained  as  follows  :  If  a  bar  of  iron  (or  bismuth)  were  magnetized  to 
the  same  degree  irrespective  of  its  direction  in  a  uniform  field,  it  would  stand  indif- 
ferently in  any  position,  but  as  a  matter  of  fact,  an  iron  rod  is  more  strongly  magnetized 
when  it  is  parallel  to  a  magnetic  field  than  when  it  is  at  right  angles  to  the  field, 
because  of  the  demagnetizing  action  of  the  free  poles  on  the  rod,  and  the  result  is  that 
the  rod  takes  up  the  position  in  which  it  is  most  strongly  magnetized.  On  the  other 
hand,  the  effect  of  the  free  magnetic  poles  on  a  rod  of  a  diamagnetic  substance  is  to 
increase  the  negative  magnetization,  so  that  the  negative  magnetization  of  a  rod  of 
bismuth  is  least  when  it  is  parallel  to  the  magnetic  field  in  which  it  is  placed.  A  rod 
of  iron  tends  to  place  itself  in  the  direction  in  which  it  is  most  strongly  magnetized  by 
the  field,  and  a  rod  of  bismuth  tends  to  place  itself  in  the  direction  in  which  it  is  least 
strongly  magnetized  by  the  field,  and  in  each  case  this  position  is  parallel  to  the  lines 
of  force  if  the  field  is  uniform. 


88  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Weber's  theory  of  diamagnetism.  — A  mass  of  copper  near  the  end  of  an  iron  rod 
has  electric  currents  induced  in  it  when  the  iron  rod  is  suddenly  magnetized,  and  as 
long  as  this  current  continues  to  flow  in  the  copper,  the  copper  is  strongly  repelled  by 
the  magnet,  the  lines  of  force  from  the  magnet  tend  to  spread  out  and  pass  around 
the  copper.  The  electrical  resistance  of  the  copper,  however,  very  soon  stops  the 
induced  current  and  then  the  strong  repulsion  ceases.  The  diamagnetic  property  of  a 
substance  has  been  explained  by  Weber  on  the  hypothesis  that  the  molecules  of  the 
substance  are  perfect  electrical  conductors  so  that  permanent  electrical  currents  are 
induced  in  the  molecules  when  the  substance  is  brought  near -a  magnet. 

PROBLEMS. 

60.  Two  permanent  magnets  I  centimeter  x  f  centimeter  X  30 
centimeters  long  are  magnetized  to  an  intensity  of  700  units  pole 
per  square  centimeter  of  sectional  area,    (a)  Calculate  the  strength 
of  each  pole,     (b)  Calculate  the  force  with  which  the  north  pole 
of  one  rod  attracts  the  south  pole  of  the  other  rod  when  the  poles 
are  at  an  approximate  distance  of  I  o  centimeters  from  each  other. 
Ans.  (a)  350  units  pole.     (£)   1,225  dynes. 

Note. — In  this  and  the  succeeding  problems  assume  the  poles  of  the  magnet  to  be 
concentrated  at  the  center  of  the  ends  of  the  bars.  The  intensity  of  magnetization  of 
an  iron  rod  is  the  strength  of  pole  on  one  end  divided  by  the  sectional  area  of  the  rod. 
See  Art.  47. 

61.  The  two  magnets  specified  in  problem  60  are  arranged  as 
shown  in  Fig.  5 1 .     Find  the  total  force  with  which  one  magnet 
acts  upon  the  other.     Ans.  —  227.39  dynes  (attraction). 


S  _  N  S  _  tf 

20cm. 


_ 


— —-.— X 


30  cm: 

Fig.  51.  Fig.  52. 

62.  The  two  magnets  specified  in  problem  60  are  arranged  as 
shown  in  Fig.  52.     Find  the  total  force  with  which  one  magnet 
acts  on  the  other.     Ans.  -f  507.8  dynes  (repulsion). 

63,  A  magnet  I  by  J  by  40  centimeters  long  having  800  units 
pole  per  square  centimeter  of  sectional  area  is  laid  across  one  of 
the  magnets  specified  in  problem  60,  as  shown  in  Fig.  53.      Find 


THE   MAGNETISM    OF    IRON. 


89 


the  total  force  with  which  one  magnet  acts  on  the  other.  Ans. 
5,376  dyne-centimeters  of  torque  tending  to  turn  magnets  as 
shown  by  arrows  in  Fig.  53. 

64.  The  two  magnets  specified  in  problem  60  are  hung  from  a 
balance  beam  as  indicated  in  Fig.  54.     Assuming  that  the  mag- 


1 


«  ^  e 
fr 

N 

r 
S 

4  „__.._     [_  —  3,         j 

Jsr 

Fig.  53. 


Fig.  54. 


30  cm. 


S 


nets  exactly  balance  each  other  before  they  are  magnetized,  find 
the  number  of  grams  which  must  be  added  to  one  pan  to  balance 
the  magnets  after  they  are  magnetized  and  specify  to  which  pan 
it  must  be  added.  Ans.  0.715  grams  must  be  added  to  the 
left  pan. 

65.  Determine  the  intensity    H  of  the   magnetic  field  at  a 
point  p   distant  1 8  centimeters  from  one  pole  and  24  centimeters 
from  the    other   pole    of  one    of  the 

magnets  specified  in  problem  60,  and 
determine  the  value  of  the  angle  6, 
as  shown  in  Fig.  55.  Ans.  //=  1.24 
gausses,  6  =  203°  46'.  5. 

66.  The   intensity    of    the   earth's 
magnetic  field  at  Washington  is  0.58 
gauss  and   its  dip  is  62°.     Find   its 
horizontal  and   vertical    components. 
V=  0.512  gauss. 

67.  Find  the  direction  and  intensity  of  the  resultant  magnetic 
field  at  a  point  30  centimeters  due  magnetic  north  of  an  isolated 


Ans.    H=  0.272  gauss, 


9o 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


north  pole  of  600  units  strength  at  Washington.  Ans.  1.07 
gausses,  north,  and  dipping  at  tan"1  0.545 2  (28° 36')  below  the 
horizontal. 

68.  Find   the   distance,  and  direction  from  the   magnet  pole 
specified  in  problem  67  to  the  point  at  which  the  resultant  field 
is  zero.     Ans.   32.16  centimeters,  south,  and  elevated  62°  above 
the  horizontal. 

69.  A  room  6  meters  long  by  5  meters  wide  by  3  meters  high 
has  its  longest  dimension  magnetic  north  and  south.     The  inten- 
sity of  the  earth's  field  in  the  room  is  0.62  gauss  and  the  dip  is 
72°.     Find  the  number  of  lines  of  magnetic  flux  across  each  of 
the  walls,  the  ceiling,  and  floor  of  the  room  and  specify  in  each 
case  whether  the  flux  is  passing  out  of  the  room  or  into  the  room. 
Ans.  East  wall,  o ;  west  wall,  o ;  north  wall,  28,740  maxwells 
out;  south  wall,  28,740  maxwells  in;  ceiling,  176,900  maxwells 
in;  floor,  176,900  maxwells  out. 

70.  The  pole  face  of  the  field  magnet  of  a  dynamo  has  an  area 
2O  centimeters  by  30  centimeters.     The  magnetic  field  between 
the  pole  faces  and  the  armature  core  is  perpendicular  to  the  pole 
face  at  each  point  and  its  intensity  is  6,000  gausses.     Calculate 
the  number  of  lines  of  force  which  pass  from  the  pole  face  into 
the  armature  core.     Ans.   3,600,000  maxwells. 

71.  Calculate  the  number  of  lines  of  force  which  emanate  from 
the  north  pole  of  one  of  the  magnets  specified  in  problem  60. 
Ans.  4,400  maxwells. 

N  N       .N       .N          N       N 

18cm. 


S 


end  view 


nide  view 


Fig.  56. 


THE   MAGNETISM    OF    IRON.  91 

72.  A  very  long  steel  ribbon  of  which  the  thickness  is  o.  I  cen- 
timeter and  the  width  is  30  centimeters  is  magnetized  so  that  one 
edge  becomes  a  north  pole  and  the  other  edge  becomes  a  south 
pole,  as  shown  in  Fig.  56,  the  intensity  of  magnetization  being 
800  units  pole  for  each  square  centimeter  of  section  of  the  steel 
(80  units  pole  for  each  centimeter  length  of  edge).      Find  the  in- 
tensity of  the  magnetic  field  due  to  the  north  polar  edge  of  the 
strip  at  a  point  distant  1 8  centimeters  from  the  edge  and  specify 
its  direction.     Ans.   8.89  gausses. 

73.  Find  the  intensity  of  the  resultant  field  at  the  point  /   in 
Fig.  56,  and  determine  the  value  of  the  angle    6,    using  the  data 
given  in  problem  72.     Ans.    H  —  ii.n  gausses,  6  =  16°  1 6'. 

74.  One  of  the  magnets  specified  in  problem  90  is  balanced 
horizontally  on  a  knife  edge  at  Washington.     The  magnet  weighs 
1 20  grams.     Find  the  horizontal  distance  from  the  knife  edge  to 
the  center  of  the  bar  taking  the  acceleration  of  gravity  to  be  980 
centimeters  per  second  per  second.     Use  the  data  specified  in 
problem  66.     Ans.   0.046  centimeter. 

75.  The  moment  of  inertia  of  one  of  the  magnets  specified  in 
problem  60  is  9,000  gr.-cm2.      Calculate  the  time  of  one  com- 
plete oscillation  of  this  magnet  when  it  is  suspended  horizontally 
at  Washington.     Ans.    11.15  seconds. 

76.  A  magnet  makes  one  complete  oscillation  per  second  in  a 
magnetic  field  of  which  the  intensity  is  0.2   gauss.      Another 
magnet  is  twice  as  long,  twice  as  wide,  and  twice  as  thick,  it  is 
magnetized  to  twice  the  intensity  (units  pole  per  units  sectional 
area)  and  it  is  suspended  in  a  field  of  which  the  intensity  is  o.i 
gauss.     What  is  its  period  of  oscillation  ?     Ans.   2  seconds. 

Note.  — The  moment  of  inertia  of  a  rotating  body  is  equal  to  the  product  of  the 
mass  of  the  body  into  the  square  of  its  radius  of  gyration.  Given  two  bodies  of  exactly 
the  same  shape,  their  radii  of  gyration  are  proportional  to  their  linear  dimensions 
whereas  their  masses  are  proportional  to  their  volumes. 

77.  A  suspended  magnet  makes  20  oscillations  in  184.5  sec~ 
onds  at  one  place,  and  20  oscillations  in  215.8  seconds  at  another 


92  ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

place.  What  is  the  ratio  of  the  intensities  of  the  horizontal  com- 
ponent of  the  earth's  magnetic  field  at  the  two  places,  and  at 
which  place  is  it  the  more  intense?  Ans.  1.367.  Field  more 
intense  at  first  place. 

78.  Two  flat -ended  poles  arranged  as  shown  in  Fig.  46  are 
observed  to  pull  towards  each  other  with  a  force  of  1,500  pounds. 
The  steel  rods  are  round  with  a  diameter  of  3  inches,     (a)  Find 
the  intensity  of  the  magnetic  field  in  the  region  between  the  flat 
poles  in  gausses,     (b)  Find  the  total  strength  of  each  pole.    Ans. 
(a)  19,170  gausses,  (&)  69,570  units  pole. 

Note.  —  Equation  (i  )  on  page  78  expresses  that  part  of  the  force  action  between 
the  two  poles  in  Fig.  46  which  depends  upon  the  polarity  of  the  rods  alone.  If 
the  field  between  the  ends  of  the  rods  is  due  in  part  to  the  direct  action  of  the 
magnetizing  coil,  then  the  force  of  attraction  between  the  two  rods  becomes 
S/STT  X  (-#i2  +  V-H^Hi  -f-  -#22)>  wnere  &\  is  tne  ^e^  due  to  the  magnetic  polarity  on 
the  ends  of  the  rods,  and  //"2  is  the  field  due  to  the  direct  action  of  the  magnetizing 
coils.  Therefore,  this  total  force  consists  of  three  parts,  namely,  sH^^ir,  2sJ7lff.J%7r, 
and  J/T^/STT.  The  first  of  these  three  parts  is  the  force  of  attraction  of  the  magnetic 
poles  on  the  ends  of  the  rods,  and  the  second  and  third  parts  are  forces  which  act  in 
part  upon  the  iron  and  in  part  upon  the  coils  of  wire  which  are  wound  upon  the  iron. 

79.  Find  the  total  magnetic  energy  in  the  room  specified  in 
problem  69.     Ans.    1,377,000  ergs. 


CHAPTER   IV. 
MAGNETIC   EFFECT   OF   THE   ELECTRIC   CURRENT.* 

50,  The  magnetic  field  due  to  an  electric  wire.  —  The  behavior 
of  a  compass  needle  in  the  neighborhood  of  an  electric  wire  shows 
that  the  region  surrounding  an  electric  wire  is  a  magnetic  field. 
The  lines  of  force  of  this  magnetic  field  encircle  the  wire.  Thus 
Fig.  57  shows  the  way  in  which  iron  filings  arrange  themselves 


Fig.  57. 

in  filaments  around  a  long  straight  electric  wire,  the  black  circle 
at  the  center  of  the  figure  represents  a  section  of  the  wire,  which 
is  perpendicular  to  the  plane  of  the  figure. 

The  lines  of  force  of  the  magnetic  field  due  to  a  circular  loop 
or  coil  of  wire  are  shown  in  Fig.  58.  In  general,  the  lines  of 
force  of  the  magnetic  field  produced  by  a  coil  of  wire  trend  in- 
wards toward  the  opening  of  the  coil  at  one  end,  pass  through 
the  opening  of  the  coil,  and  spread  out  at  the  other  end.  Thus, 

*  Chapter  V  on  Induced  Electromotive  Force,  and  Chapter  VI  on  Inductance  con- 
stitute continuations  of  this  general  subject,  the  Magnetic  Effect  of  the  Electric 
Current. 

93 


94 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


a  long  coil  of  wire  is  exactly  equivalent  to  a  magnet  in  so  far  as 
its  relation  to  surrounding  objects  is  concerned,  lines  of  magnetic 
force  flow  out  of  one  end  of  the  coil  through  the  surrounding 

<»  region  and  into  the  other  end 
of  the  coil  in  the  same  way 
that  lines  of  force  flow  out 
from  the  north  pole  of  a  mag- 
net through  the  surrounding 
region  and  in  towards  the 
south  pole  of  the  magnet. 

The  behavior  of  a  magnet  in 
.A.XJS_OF_CPJL_ tjie    neighborhood  of    a    long 

straight  electric  wire.  —  The 
small  circles  in  Figs.  59  and 
60  represent  the  section  of  a 
long  straight  wire  in  which 
current  is  flowing  towards* 
the  reader.  Figure  59  shows 
the  forces  .Wwith  which  the 
magnetic  field  due  to  the  wire 
acts  on  the  poles  of  a  moder- 
ately long  magnet,  and  Fig.  60  shows  the  forces  FF  with  which 
the  magnetic  field  of  the  wire  acts  upon  the  poles  of  a  very  short 
magnet.  Thus,  a  long  magnet  is  drawn  towards  the  wire, 
although  the  forces  acting  on  each  pole  are  at  right  angles  to  the 
dotted  lines  in  Fig.  59,  whereas  a  very  short  magnet  is  not  per- 
ceptibly attracted  by  the  wire  because  the  two  forces  FF  in 
Fig.  60  are  very  nearly  opposite  to  each  other  in  direction.  The 
north  pole  of  a  magnet  tends  to  move  around  the  wire  in  one 
direction  and  the  south  pole  of  a  magnet  tends  to  move  around 
the  wire  in  the  opposite  direction.  Thus,  the  north  pole  tends  to 

*  In  representing  a  flow  of  current  towards  the  reader  in  the  section  of  a  wire,  a 
dot  is  used  as  if  one  were  looking  at  the  point  of  an  arrow,  and,  when  representing  a 
flow  of  current  away  from  the  reader,  a  cross  is  used  as  if  one  were  looking  at  the 
feathered  end  of  an  arrow  ;  thus,  O  represents  a  flow  of  current  towards  the  reader 
and  ©  represents  a  flow  of  current  away  from  the  reader. 


/ 


Fig.  58. 


MAGNETIC    EFFECT   OF    ELECTRIC    CURRENT. 


95 


move  around  the  wire  in  a  counter-clockwise  direction  in  Figs. 
59  and  60.  The  direction  of  a  current  in  a  wire  may  be  deter- 
mined by  means  of  the  compass,  as  follows  :  Bring  the  compass 


N 


Fig.  59. 


•F      S  N  F 

Fig.  60. 


near  the  wire,  and,  knowing  that  the  forces  which  act  on  the  two 
poles  of  the  compass  needle  are  at  right  angles  to  lines  drawn  from 
the  poles  to  the  wire,  infer,  from  the  observed  movements  of  the 
needle,  the  direction  in  which  the  north-pointing  pole  of  the  nee- 
dle tends  to  move  around  the  wire.  The  direction  of  the  current 
in  the  wire  *  is  the  direction  in  which  a  right-handed  screw  with  its 
axis  parallel  to  the  wire  would  travel  if  the  screw  is  turned  in  the 
direction  in  which  the  north-pointing  pole  tends  to  move  around  the 
wire. 

51.  The  composite  magnetic  field  which   is  produced  when  a 
straight  electric  wire  is  stretched  across  a  region  which,  but  for  the 


,/'/ '•S'  '' •'r*^'      ^-~ >r~~^L~^.  •    >*."     -     "^ »  v >T -X  *C'\ 


*  See  Art.  2. 


Fig.  61. 


96 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


presence  of  the  electric  wire,  would  be  a  uniform  field.  —  The  mag- 
netic field  between  the  flat-ended  magnet  poles  in  Fig.  48  is  sensi- 
bly uniform.  Figure  6 1  shows  the  same  field  modified  by  the  pres- 
ence of  a  straight  electric  wire.  /The  small  black  circle  in  Fig.  6 1 
represents  the  section  of  the  wire  and  the  wire  is  perpendicular  to 
the  plane  of  the  figure.  The  magnetic  field  in  Fig.  61  is  due  to 
two  distinct  causes,  namely,  (a)  the  two  flat-eiided  magnet  poles, 
and  (b)  the  electric  wire,  and  it  may  therefore  be  called  a  com- 
posite field.  If  the  field  were  due  to  the  wire  alone  its  lines  of 


•'••"-^ 


Fig.  62. 

force  would  be  circles  as  shown  in  Fig.  57,  and  if  the  field  were 
due  to  the  flat-ended  poles  alone  its  lines  of  force  would  be  as 
shown  in  Fig.  48.  The  trend  of  the  lines  of  force  in  Fig.  61  in 
the  immediate  neighborhood  of  the  wire  are  more  clearly  shown 
in  Fig.  62  which  is  from  a  drawing. 

Side  push  on  an  electric  wire  which  is  stretched  across  a  uniform 
magnetic  field. — The  wire  shown  in  Figs.  61  and  62  is  pushed 
sidewise  by  the  magnetic  field  *  as  indicated  by  the  arrow  F  in 

*  Strictly,  one  should  perhaps  speak  of  the  side  force  on  the  wire  in  Figs.  6 1  and 
62  as  due  to  the  two  magnet  poles,  because  the  two  magnet  poles  constitute  the  actual 


MAGNETIC   EFFECT   OF   ELECTRIC   CURRENT. 


97 


Fig.  62.  This  side  force  is  at  right  angles  to  the  wire  and  to  the 
magnetic  field  (Fig.  48)  which  is  acting  on  the  wire.  The  side 
force  which  acts  upon  the  wire  in  Fig.  6 1  may  be  ascribed  to  the 
tension  of  the  lines  of  force. 

Examples.  —  The  simple  example  of  the  magnetic  effect  of  the 
electric  current  which  is  cited  in  Art.  I  and  represented  in  Fig.  I 
illustrates  the  side  push  of  a  magnetic  field  on  a  wire  inasmuch 
as  the  magnetic  field  which  emanates  from  the  north  pole  of  the 
magnet  in  Fig.  I  is  partly,  at  least,  at  right  angles  to  the  wire 
AB.  The  side  push  on  an  electric  wire  in  a  magnetic  field  is  also 
exemplified  in  the  electric  motor.  A  cylindrical  mass  of  iron  A, 
Fig.  63,  has  wires  arranged  on  its  surface  parallel  to  its  axis,  and 


Fig.  63. 

the  whole  is  placed  between  two  magnet  poles  NS,  as  shown. 
The  narrow  region  between  the  cylinder  A  and  the  poles  N  and 
5  is  a  strong  magnetic  field,  the  lines  of  force  of  which  are  radial. 
An  electric  current  is  made  to  flow  through  the  wires  in  the  direc- 
tions indicated  by  the  dots  and  crosses,  and  the  result  is  that  the 

visible  agent  which  is  acting  on  the  wire.     It  is  very  important,  however,  that  the 
student  become  familial  with  the  idea  of  a  magnetic  field  as  a  physical  reality,  and  to 
ascribe  the  side  force  in  Figs   6 1  and  62  to  the  field  which  is  produced  by  the  two  mag- 
net poles  puts  the  whole  matter  in  the  most  intelligible  form. 
8 


98 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


wires  are  pushed  sidewise  by  the  magnetic  field  causing  the 
cylinder  A  to  rotate  in  the  direction  of  the  curved  arrow. 

52,  Strength  of  current  magnetically  defined.  —  Consider  a 
straight  electric  wire  stretched  'across  a  uniform  magnetic  field 
of  which  the  intensity  is  one  gauss,  the  wire  being  at  right  angles 
to  the  field  as  described  in  the  foregoing  article.  The  force  in 
dynes  with  which  the  field  pushes  sidewise  on  one  centimeter  of 
this  wire  has  been  adopted  as  the  fundamental  measure  of  the 
strength  of  the  current  in  the  wire.  This  force-per-unit-length-of- 
wire-per-unit-field-intensity  is  called  simply  the  strength  of  the 
current  in  the  wire ;  let  it  be  represented  by  /.  The  force  push- 
ing sidewise  on  /  centimeters  of  the  wire  is  //,  and,  if  the  field 
intensity  is  H  gausses  instead  of  one  gauss,  the  force  is  H 
times  as  great,  or  IIH\  that  is, 

F=IIH  (28) 

in  which  F  is  the  force  in  dynes  pushing  sidewise  upon  /  centi- 
meters of  wire  at  right  angles  to  a  uniform  magnetic  field  of  which 
the  intensity  is  H  gausses,  and  /  is  the  strength  of  the  current 
in  the  wire. 


Definition  of  the  ampere. — The  ampere 
is  one  tenth  of  an  abampere. 


Definition  of  the  eibampere.  —  A  wire 
is  said  to  carry  a  current  of  one  abampere 
when  one  centimeter  of  the  wire  is  pushed 
sidewise  with  a  force  of  one  dyne  when  the 
wire  is  stretched  across  a  magnetic  field  of 
which  the  intensity  is  one  gauss.  That  is, 
F  in  equation  (28)  is  expressed  in  dynes 
when  /  is  expressed  in  centimeters,  H  in 
gausses,  and  /  in  abamperes. 


In  the  early  days  of  the  development  of  the  theory  of  electricity  and  magnetism, 
a  great  variety  of  arbitrary  units  was  used.  Thus,  the  resistance  of  a  particular  piece 
of  wire  would  be  used  as  a  unit  of  resistance,  the  electromotive  force  of  a  particular 
voltaic  cell  would  be  used  as  a  unit  of  electromotive  force,  and  current  values  were 
often  specified  in  terms  of  the  deflections  of  a  particular  galvanometer.  The  introduc- 
tion of  a  uniform  system  of  units  was  due  chiefly  to  Weber  and  Gauss  in  Germany  and 
to  Maxwell  and  Kelvin  in  England.  This  uniform  system  of  units  was  based  on  the 
units  already  in  use  in  mechanics,  the  centimeter,  the  gram,  and  the  second,  and  the 
units  of  this  c.g.s.  system  were  called  absolute  units. 


MAGNETIC   EFFECT   OF   ELECTRIC   CURRENT. 


99 


The  electrical  units  which  are  now  almost  universally  employed,  the  ampere,  the 
ohm,  the  volt,  the  coulomb,  the  henry,  and  the  farad  are,  however,  not  the  original 
c.g.s.  units,  but  multiples  or  submultiples  of  them  The  original  c.g  s.  units  as  a  rule 
have  no  names.  Therefore  in  this  text  the  c.g.s.  units  (of  the  so-called  "electromag- 
netic" system)  which  correspond  to  the  ampere,  the  ohm,  the  volt,  etc.,  are  desig- 
nated by  the  prefix  ab.  Thus,  we  have  the  abampere,  the  abohm,  the  abvolt,  etc. 


Definition  of  the  abohm. — A  wire  has  a 
resistance  of  one  abohm  when  one  erg  of 
heat  is  generated  in  it  in  one  second  by 
a  current  of  one  abampere.  When  H  in 
equation  (2),  Art.  12,  is  expressed  in  ergs, 
/  in  seconds,  and  /  in  abamperes,  then 
R  is  expressed  in  abohms. 

Definition  of  the  abvolt.  — An  electric 
generator  has  an  electromotive  forceof  one 
abvolt  when  it  delivers  one  erg  per  sec- 
ond of  power  with  a  current  output  of  one 
abampere  [see  equation  (6),  Art.  18]. 

The  abvolt  may  be  defined,  on  the  basis 
of  Ohm's  Law,  as  an  electromotive  force 
which  is  capable  of  producing  a  current  of 
one  abampere  through  a  circuit  of  which 
the  resistance  is  one  abohm. 


Definition  of  the  ohm.  — A  wire  has  a 
resistance  of  one  ohm  when  one  joule  of 
heat  is  generated  in  it  in  one  second  by  a 
current  of  one  ampere.  The  ohm  is  equal 
to  io9  abohms. 


Definition  of  the  volt. — An  electric  gen- 
erator has  an  electromotive  force  of  one 
volt  when  it  delivers  one  joule  per  second 
(one  watt)  of  power  with  a  current  output 
of  one  ampere.  The  volt  is  equal  to  io8 
abvolts. 

The  volt  may  be  defined,  on  the  basis 
of  Ohm's  Law,  as  an  electromotive  force 
which  is  capable  of  producing  a  current  of 
one  ampere  through  a  circuit  of  which  the 
resistance  is  one  ohm. 


Side  force  on  an  electric  wire  which  is  not  at  right  angles  to  a 
magnetic  field. — When  an  electric  wire  is  parallel  to  a  magnetic 
field,  no  force  acts  on  the  wire.  If  the  angle  between  the  wire 
and  the  direction  of  the  field  is  6,  then  the  field  may  be  resolved 
into  two  components  //"sin  Q  and  //cos  0,  perpendicular  to 
and  parallel  to  the  wire,  respectively;  the  latter  component  has 
no  action  on  the  wire  and  the  former  component  produces  the 
side  force 

F=Itffsm0  (29) 

If  the  wire  is  not  straight,  or  if  the  field  is  not  uniform,  then  one  must  consider  the 
force  action  on  an  element  of  the  wire,  and  equation  (29)  becomes 

A^=/^sin0.A/  (30) 

in  which  A/  is  a  short  portion,  or  element,  of  the  wire,  H  is  the  intensity  of  the  field 
at  the  element,  6  is  the  angle  between  If  and  A/,  /  is  the  strength  of  the  current 
in  the  wire  in  abamperes,  and  A^  is  the  force  pushing  on  A/.  This  force  is  perpen- 
dicular both  to  H  and  to  A/. 


100        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


53.  Contribution  to  the  magnetic  field  at  a  given  point  by  one  element  of  an 
electric  wire.  —  The  region  surrounding  an  electric  circuit  is  a  magnetic  field  and 
each  element  of  the  wire  which  constitutes  the  circuit  may  be  considered  as  contribu- 
ting its  share  to  the  field  intensity  at  each  point.  Imagine  a  magnet  pole  of  strength 
m  to  be  placed  at  the  point  at  which  it  is  desired  to  find  the  field  intensity  kH  which 
is  produced  by  a  given  element  A/  of  the  wire. 

Let  r  be  the  distance  from  m  to  A/  and 
let  6  be  the  angle  between  r  and  A/,  as 
shown  in  Fig.  64.  'The  field  intensity  at  the 
element  due  to  the  pole  is  mjr"*  according  to 
equation  (17).  The  component  of  this  field 
which  is  at  right  angles  to  the  element  is 
w/r2Xsin  0»  and  this  component  of  the 
field  pushes  sidewise  on  the  wire  with  a  force 
which  is  given  by 


. 


Fig 


according  to  equation  (30).  This  is  the 
force  with  which  the  pole  m  acts  on  the 
element,  and  therefore  it  is  also  the  force 
(disregarding  sign)  with  which  the  element 
acts  upon  the  pole.  But  the  force  with  which  the  element  acts  upon  the  pole  must  be 
equal  to  the  product  of  the  strength  of  the  pole  and  the  field  intensity  at  the  pole  due 
to  the  element,  that  is, 


whence 


1  sin  6 


(30 


in  which    A//"  is  the  field  intensity  at  the  point   m    in  Fig.  64  due  to  the  element 
A/.     This  field    A//"  is  perpendicular  to    r   and  to    A/. 

Note.  —  It  is  evident  from  the  above  discussion  that  the  magnetic  field  at  a  given 
point  in  the  neighborhood  of  a  given  coil  of  wire,  or  a  circuit  of  any  form,  in  which 
an  electric  current  is  flowing  is  proportional  to  the  strength  of  the  current,  and  that 
its  direction  is  fixed.  That  is  to  say,  if  the  strength  of  the  current  is  doubled  the  field 
intensity  is  doubled  everywhere,  but  the  direction  of  the  field  is  everywhere  unaltered. 
The  trend  of  the  lines  of  force  of  the  magnetic  field  due  to  a  given  coil  or  circuit  de- 
pends only  upon  the  shape  and  size  of  the  coil. 

54.  The  intensity  of  the  magnetic  field  at  the  center  of  a  circu- 
lar loop  of  wire.  —  If  we  can  calculate  the  force  with  which  a  cir- 
cular loop  of  wire  with  given  current  acts  on  a  magnet  pole  of 
given  strength  placed  at  the  center  of  the  circular  loop,  we  can 
derive  an  expression  for  the  intensity  of  the  field  at  the  center 
of  the  loop  due  to  the  current,  because  the  force  exerted  on  the 


MAGNETIC    EFFECT   OF    ELECTRIC    CURRENT. 


IOI 


pole  by  the  loop  of  wire  must  be  equal  to  the  intensity  of  the 
field  at  the  pole  due  to  the  loop  multiplied  by  the  'strength  of  the 
pole  according  to  equation  (16).  Consider  therefore  a  magnet 
pole  of  strength  m  placed  at  the  center  of  the  circular  loop  as 
shown  in  Fig.  65.  This  pole  produces  a  magnetic  field  of  which 
the  intensity  at  the  wire  is 
mfr*,  and  which  is  every- 
where at  right  angles  to 
the  wire.  Therefore  the 
force  with  which  the  wire 
is  pushed  sidewise  (per- 
pendicular to  the  plane  of 
the  paper  in  Fig.  65)  is 
equal  to  the  product  of  the 
length  of  the  wire,  the  in- 
tensity of  the  field  (m/r2), 
and  the  strength  of  the 
current  /  in  the  wire  in  abamperes ;  but  the  length  of  the  wire  is 
2irrZ  where  Z  is  the  number  of  turns  of  wire  in  the  loop,  so  that 
2irrZ  x  mjr2  x  /  is  the  force  with  which  the  wire  is  pushed 
sidewise  by  the  pole  m. '  But,  disregarding  sign,  this  is  equal  to 
the  force  mH  with  which  the  loop  of  wire  pushes  on  the  pole. 
Therefore  we  have 


Fig.  65. 


m 


2irrZ  x  -5  X  / 


from  which  we  obtain 


H. 


2irZI 


(32) 


55.  Magnetic  field  in  the  neighborhood  of  a  long  straight  electric  wire.  — 

The  lines  of  force  of  the  magnetic  field  surrounding  a  long  straight  electric  wire  are 
circles  with  their  planes  at  right  angles  to  the  wire  and  their  centers  on  the  axis  of  the 
wire,  as  explained  in  Art.  50  and  as  shown  in  Fig.  57.  To  derive  an  expression  for 
the  intensity  of  this  field  at  a  point  distant  r  centimeters  from  the  axis  of  the  wire, 
proceed  as  follows  :  A  long  straight  wire  AB  carries  a  current  of  7  abamperes, 
and  a  long  magnetized  steel  strip  is  placed  with  its  north  polar  edge  parallel  to  AB 
and  at  a  distance  of  r  centimeters  from  AB  as  shown  in  Fig  66.  The  magnetic 
field  due  to  AB  has  the  same  value  all  the  way  along  the  polar  edge  NNNN,  as 
is  evident  from  considerations  of  symmetry,  the  wire  being  indefinitely  long.  Consider 


102        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


magnetized  strip 
of  steel 


s         s 

side  view 


S  S 

Fig.  66. 


end  view 


a  portion  of  the  polar  edge  NNNN  and  a  portion  of  the  wire  AB  each  /  centi- 
meters in  length.  The  intensity  H'  of  the  magnetic  field  at  the  wire  due  to  the  pole 
NNNN  is  equal  to  zmjrl,  according  to  equation  (20),  and  the  wire  AB  is  pushed 
sidewise  by  a  force  F/  which  is  equal  to  iy^iy^2mjrlt  according  to  equation 
(28),  but  the  force  with  which  the  pole  NNNN  acts  on  the  wire  is  equal  and  oppo- 
site to  the  force  mH  (see  Fig.  66)  with  which  the  wire  acts  on  the  pole.  Therefore, 
ignoring  signs,  we  have  mH=.Iiy(2mlrl,  whence 

ff=^  (33) 

56,  Magnetic  field  inside  of  a  long  solenoid.  —  A  solenoid  is  a 
winding  of  wire  on  a  long  tube  as  shown  in  Fig.  67,  which  is  a 
sectional  view.  When  an  electric  current  flows  through  the  wind- 
ing of  a  solenoid  the  region  inside  of  the  solenoid  becomes  a 
uniform  magnetic  field  except  near  the  ends  of  the  solenoid  as 


Fig.  67. 


MAGNETIC   EFFECT   OF   ELECTRIC   CURRENT.         103 

shown  by  the  fine  lines  in  Fig.  67,  and  the  intensity  of  this  field 
is  given  by  the  equation 

H^qirzl  (34) 

in  which  H  is  the  intensity  of  the  field  in  gausses,  z  is  the  num- 
ber of  turns  of  wire  on  each  centimeter  of  length  of  the  solenoid, 
and  /  is  the  strength  of  the  current  in  abamperes.  If  the  current 
is  expressed  in  amperes  equation  (34)  becomes 


=  ^--*7 
10 


(35) 


in  which  H,   as  before,  is  expressed  in  gausses. 

In  order  to  derive  equation  (34)  let  us  consider  the  arrange- 
ment shown  in  Fig.  68  consisting  of  a  long  coil  having  z  turns 


long  coil 

GQ©00i 


feel  rod 


Ifpn? 

111  I  I  I  \\poleoft 


pole  of  strength  m 


WttffiW^^  *      > 


side  view  (section) 

Fig.  68. 


long  coil 


end  view 


of  wire  on  each  centimeter  of  its  length,  with  a  steel  rod  pro- 
jecting into  it.  Let  us  assume  that  the  total  pole  strength  m  on  the 
end  of  the  steel  rod  is  spread  uniformly  *  over  a  portion  of  the  rod 
of  length  /,  as  indicated  by  the  shading  in  Fig.  68.  The  lines  of 
force  emanate  from  such  a  long  pole  in  planes  at  right  angles  to 
its  length  as  shown  in  Fig.  68,  and  the  intensity  of  this  field  at 
the  surface  of  the  long  coil  is  Hf=  2mjrl,  according  to  Art.  40, 
where  r  is  the  radius  of  the  solenoid  and  /  is  the  length  in  centi- 

*  It  would  be  very  difficult  indeed  to  magnetize  a  rod  so  as  to  have  its  pole  spread 
uniformly  over  a  given  length  of  the  end  of  the  rod,  especially  when  the  rod  projects 
into  a  solenoid  as  shown  in  Fig.  68,  because  the  effect  of  the  solenoid  is  to  tend  to 
concentrate  the  magnet  pole  at  the  end  of  the  rod.  The  assumed  distribution  of  pole 
is,  however,  a  possibility  if  the  current  in  the  solenoid  is  very  weak  and  therefore  the 
assumed  distribution  is  a  legitimate  basis  for  the  discussion  of  equation  (34). 


104        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

meters  of  the  portion  of  the  steel  rod  over  which  the  pole  is  uni- 
formly distributed.  The  non-uniformity  of  the  field  near  the 
ends  of  the  slim  pole  is  negligible  if  r  is  small  in  comparison 
with  /.  The  effect  of  this  slim  pole  is  therefore  to  produce  a 
radial  magnetic  field  over  the  whole  of  a  portion  of  the  coil  of 
length  /.  This  portion  of  the  coil  contains  Iz  turns  of  wire,  and 
the  length  of  each  turn  is  2irr  so  that  the  total  length  of  wire  in 
the  region  where  field  is  produced  by  the  slim  pole  is  2frrlz. 
This  wire  is  everywhere  at  right  angles  to  the  field  H1  (which  is 
due  to  the  slim  pole)  and  it  is  therefore  pushed  sidewise  by  a  force 
F=  /x  2irrlz  x  H' y  or,  using  2mjrl  for  //',  we  have 


but  the  force  with  which  the  slim  pole  pushes  on  the  coil  is  equal 
and  opposite  to  the  force  with  which  the  coil  pushes  on  the  pole, 
and  the  force  with  which  the  coil  pushes  on  the  pole  is  equal  to 
the  product  of  the  strength  of  the  pole  and  the  field  intensity  at 

the  pole  due  to  the  coil.  There- 
fore the  field  intensity  inside  of 
the  coil  is  equal  to 


57.  The  tangent  galvanometer. 
—  One  of  the  earliest  forms  of 
instrument  for  measuring  the 
strength  of  the  electric  current 
was  the  tangent  galvanometer. 
It  consists  essentially  of  a  circu- 
lar coil  of  wire  at  the  center  of 
which  a  small  magnet  is  sus- 
pended, as  shown  in  Fig.  6ga. 
This  suspended  magnet  carries  a 
pointer  which  plays  over  a  di- 
vided circle  by  means  of  which  the  angle  through  which  the 
magnet  is  turned  when  a  current  is  sent  through  the  wire  may 
be  observed.  The  coil  of  wire  is  mounted  with  its  plane  vertical 
and  magnetic  north  and  south. 


Fig.  69a. 


MAGNETIC    EFFECT   OF    ELECTRIC    CURRENT. 


105 


When  no  current  flows  through  the  coil  the  suspended  magnet 
points  in  the  direction  of  the  earth's  horizontal  field  H' .  A  cur- 
rent of  /  abamperes  in  the  coil  produces  a  magnetic  field  of  which 
the  intensity  at  the  center  of  the  coil  is 
H  =  2r7rZfjr  and  of  which  the  direction  at 
the  center  of  the  coil  is  at  right  angles  to 
Hf.  This  field  H  combines  with  H'  to 
give  a  resultant  field  R,  Fig.  69^,  in  the 
direction  of  which  the  suspended  magnet 
now  points,  <f>  being  the  angle  through  which 
the  magnet  is  turned  by  the  current.  From 
Fig.  69$  we  have 

H 
tan  <£  =  -jrf 

or,  substituting  2irZIjr  for  H  and  solving  for  /,  we  have 

rH' 

I  =  — ~  •  tan  9 
2TrZ 

This  equation  gives  the  value  of  /  in  abamperes  when  r  is  in 
centimeters  and  H  is  in  gausses,  the  values  of  r,  //',*  and  Z 
being  known  and  (f>  being  observed. 

If  /  be  expressed  in  amperes,  then  equation  (360)  becomes 


amp. 


•rrZ 


tan 


(366) 


A  serious  fault  in  the  tangent  galvanometer  is  that  the  earth's 
horizontal  field  H1  is  never  known  accurately  because  it  is  con- 
tinually changing  in  value.  When  it  is  desired  merely  to  measure 
the  ratio  of  two  currents,  however,, the  value  of  H'  need  not  be 
known  (provided  it  does  not  change  while  the  following  obser- 
vations are  being  taken).  One  current  7j  is  sent  through  the 
galvanometer,  and  the  corresponding  deflection  <^1  is  observed, 
giving 


*See  Art.  42  and  Chapter  X. 


106        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

rH' 


Then  the  other  curreat  72  is  sent  through  the  galvanometer  and 
the  corresponding  deflection    <£2   is  observed,  giving 

rH' 


Dividing  equations  (i)  and  (ii),  member  by  member,  we  have 


Figure  70  is  a  general  view  of  a  tangent  galvanometer.     The 
divisions  on  the  large  horizontal  circle  are  not  used. 


Fig.  70. 

58.  The  action  of  a  uniform  magnetic  field  upon  a  suspended 
coil  in  which  an  electric  current  is  flowing,  (a)  Simple  case  of  a 
rectangular  coil  with  two  of  its  edges  parallel  to  the  field. — Figure  7 1 
represents  a  rectangular  coil  of  wire  suspended  between  the  poles 
N  and  5  of  a  large  magnet.  Let  H  be  the  intensity  of  the 
magnetic  field  (assumed  to  be  uniform),  let  b  be  the  breadth  of 
the  coil,  let  a  be  the  height  of  the  coil,  and  let  Z  be  the 


MAGNETIC   EFFECT   OF   ELECTRIC   CURRENT.         IO/ 

number  of  turns  of  wire  in  the  coil.  The  field  H  is  parallel  to 
the  top  and  bottom  edges,  or  limbs,  of  the  coil,  and  at  right 
angles  to  the  two  side  limbs  of  the  coil.  The  right-handed  limb 
of  the  coil  in  Fig.  7 1  is  pushed  forwards  (towards  the  reader)  and 
the  left-handed  limb  of  the  coil  in  Fig.  71  is  pushed  backwards 


side  view. 


top  view 


Fig.  71. 


(away  from  the  reader),  and  the  force  in  each  case  is  equal  to 
Za  x  //X  /,  according  to  equation  (28),  /  being  the  strength 
of  the  current  in  the  coil  in  abamperes.  It  is  evident  that  the 
total  force  action  on  the  coil  is  a  torque  tending  to  turn  the  coil 
about  the  axis  of  suspension ;  the  value  of  the  torque  may  be 
obtained  by  multiplying  the  force  acting  on  each  limb  of  the  coil 
by  its  lever  arm  bJ2  and  adding  the  two  results  together,  which 

gives 

T=  abZIH  (38) 

in  which  T  is  the  torque  in  dyne-centimeters  tending  to  turn  the 
coil  and  /  is  the  current  in  the  coil  in  abamperes. 

If  the  rectangular  coil  is  allowed  to  turn  through  an  angle  6 
about  the  axis  of  suspension  in  Fig.  7 1 ,  then  only  the  component, 
H  cos  6,  of  the  field  will  be  effective  in  producing  torque  as 
shown  in  Fig.  72,  and  equation  (38)  will  become 


T=abZIHcos6 


(39) 


108        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

When  the  angle  6  is  equal  to  90°  (plane  of  coil  at  right  angles 
to  the  field  //),  then  the  torque  is  equal  to  zero. 


(6}  Case  of  a  circular  coil  -with  its  plane  parallel  to  the  magnetic  field  as  shorvn 
in  Fig.  73.  — In  this  case  let  us  consider  a  single  turn  of  the  coil  of  which  the  radius 
is  r.  The  vertical  dotted  line  in  Fig.  74  represents  the  axis  about  which  the  torque  is 
to  be  determined.  Consider  an  element  A/  of  the  wire.  The  component  of  H 
which  is  at  right  angles  to  A/  is  equal  to  If  cos  0,  the  product  H  cos  <j>  X  ^X  A/ 
gives  the  force  pushing  forwards  (towards  the  reader  in  Fig.  74)  on  the  element  A/, 


two  fine  suspending 
wires 


Fig.  73. 

and  the  product  of  this  force  times  the  lever  arm  r  sin  0  gives  the  torque  action  on 
the  element  A/,  that  is 

A  T=  tfcos  0  X  /X  r  sin  0  X  A/ 

but  cos  <j>  X  A/  is  equal  to  the  vertical  height  h  shown  in  Fig.  74,  so  that 
r  sin  $  X  cos  0  •  A/  is  equal  to  the  shaded  area  shown  in  the  figure.  Therefore,  rep- 
resenting this  shaded  area  by  A  A,  we  have 


MAGNETIC    EFFECT   OF   ELECTRIC   CURRENT. 


109 


and,  since  this  relation  is  true  for  every  element  of  the  circular  coti,  it  follows  that 
the  total  torque  is  equal  to  IH  times  the  total  area  A  enclosed  by  the  turn  of  wire, 

that  is 

T^AIH  (40) 

in  which  T  is  in  dyne-centimeters,  /  is  in  abamperes,  and  A  is  in  square  centi- 
meters, H  being  expressed  in  gausses.  If  the  coil  has  more  than  one  turn  of  wire,  A  is 
equal  to  the  sum  of  the  areas  of  all  the  turns.  Thus,  if  the  coil  has  four  turns  of  wire 
of  which  the  radii  are  a,  b,  c  and  d  respectively  then  A  =  tra2  -J-  irb2  -\-  ire*  4-  ird2. 


Fig.  74. 

59.  The  electrodynamometer  is  an  instrument  for  determining 
strength  of  an  electric  current  from  a  measurement  of  the  mutual 
force  action  between  two  coils  of  wire  through  both  of  which  the 
current  flows.  One  of  these  coils  is  fixed  and  the  other  is 
suspended.  The  magnetic  field  produced  by  the  fixed  coil  exerts 
a  force  upon  the  suspended  coil,  and  this  force,  or  the  movement 
which  it  produces,  is  observed.  When  the  coils  are  very  simple 
in  shape  it  is  possible  to  calculate  (from  geometrical  and  me- 
chanical data  alone)  the  force  action  between  the  two  coils  for 
a  given  current,  or,  conversely,  to  calculate  the  value  of  the  cur- 
rent when  the  force  action  is  observed  in  mechanical  units. 
Such  an  electrodynamometer  is  called  an  absolute  electrodyna- 
mometer. 


110        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  simplest  absolute  electrodynamometer  is  that  devised  by  Wilhelm  Weber  in 
1846.  It  consists  of  a  large  stationary  coil  mounted  with  its  plane  vertical,  and  a 
small  circular  coil  suspended  at  its  center  by  two  fine  wires.  The  current  /  to  be 
measured  flows  through  both  coils.  The  magnetic  field  produced  by  the  outer  coil  at 


Fig.  75. 

its  center  is  H=  27rZ///r/,  where  Z'  is  the  number  of  turns  of  wire  in  the  large 
coil  and  r*  is  its  radius.  This  magnetic  field  exerts  a  torque  T=  7rr//2Z/////cos  0 
upon  the  small  coil,  where  Z"  is  the  number  of  turns  of  wire  in  the  small  coil,  r"  is 
its  radius  and  6  is  the  angle  between  H  and  the  plane  of  the  small  coil  or,  in  other 
words,  6  is  the  complement  of  the  angle  between  the  planes  of  the  two  coils.  Sub- 
stituting the  value  of  ,#"=27rZ///r/  in  the  expression  for  T,  we  have 

cosfl  (41) 

This  equation  permits  the  calculation  of  /  when  Z',  Z",  r*t  and  r"  are  known, 
and  when  T  and  6  have  been  observed.  When  c.g.s.  units  are  used  in  equation 
(41),  the  current  is  given  in  abamperes. 

Figure  75  shows  a  slightly  modified  form  of  Weber's  absolute  electrodynamometer 
in  which  the  small  coil  is  suspended  in  the  approximately  uniform  field  between  two 
large  circular  coils  side  by  side. 

The  Siemens  electrodynamometer.  —  The  force  action  between 
two  coils  is  proportional  strictly  to  the  square  of  the  current 
which  flows  through  the  two  coils  whatever  the  shape  and  rela- 
tive position  of  the  two  coils  may  be,  provided  only  that  the 
relative  position  of  the  two  coils  does  not  change.  Therefore,  if 


MAGNETIC   EFFECT   OF   ELECTRIC   CURRENT.         Ill 


the  force  action  between  the  coils  is  measured,  first  for  a  current 
/'  and  then  for  a  current  /" ',  the  ratio  I'  \I'f  is  equal  to  the 
square  root  of  the  ratio  of  the  observed  force  actions.  The  elec- 
trodynamometer  of  Siemens  is  used  for  measuring  current  ratios 
in  this  way.  A  general  view  of  this  instrument  is  shown  in  Fig. 
?6a.  The  movable  coil  B  (see  Fig.  76^)  is  suspended  by  a  fine 


Fig.  76a. 


Fig.  76b. 


thread  and  its  terminals  dip  into  two  mercury  cups  which  permit 
of  its  being  connected  in  series  with  the  fixed  coil  A.  When 
current  is  allowed  to  flow  through  the  two  coils  in  series,  a  torque 
is  exerted  upon  the  movable  coil  by  the  fixed  coil,  and  the  helical 
spring  is  twisted,  by  turning  the  milled  head,  until  the  movable 
coil  is  brought  to  its  standard  position,  as  indicated  by  the 
pointer  which  is  attached  to  B.  The  angle  through  which  the 
milled  head  is  turned  is  indicated  by  the  pointer  which  is  at- 
tached to  the  milled  head,  and  this  angle  is  a  measure  of  the 
force  action  between  the  coils  so  that  this  angle  is  proportional 
to  the  square  of  the  current,  or  the  current  is  proportional  to  the 
square  root  of  the  angle. 


112        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

60.  The  sensitive  galvanometer  (Kelvin  type).  —  From  the  equa- 
tion of  the  tangent  galvanometer,  equation  (36),  it  is  evident  that 
a  given  current  will  produce  the  greatest  deflection  cf>  of  the 
galvanometer  needle  when  thtf*  number  of  turns  of  wire  in  the 
coil  is  great,  when  the  radius  of  the  coil  is  small,  and  when  the 
directing  field  H'  is  weak.  A  galvanometer  constructed  so  as 
to  meet  these  conditions  and  thus  give  a  perceptible  deflection 
for  a  very  small  current  is  called  a  sensitive  galvanometer.  Such 
a  galvanometer  is  used  chiefly  for  merely  detecting  the  presence 
of  current  in  a  circuit.  The  magnet  of  such  a  galvanometer  is 
usually  suspended  by  means  of  a  fiber  of  unspun  silk  or  quartz, 
and,  in  order  that  small  deflections  may  be  easily  detected,  a 
mirror  is  usually  attached  to  the  suspended  magnet  so  that  the 
angular  movement  of  the  suspended  magnet  may  be  observed  by 
means  of  a  telescope  and  scale. 

Use  of  a  governing  magnet.  — In  order  to  secure  a  weak 
directing  field  Hf,  the  earth's  field  may  be  partially  neu- 
tralized in  the  neighborhood  of  the  suspended  magnetic 
needle  by  properly  placing  a  large  magnet  in  the  neighbor- 
hood of  a  galvanometer.  This  large  magnet  is  called  a 
governing  magnet. 

Use  of  an  astatic  system  of  magnetic  needles. — Two  similar 
magnetic  needles  NS  and  SN  attached  to  a  rod,  as  shown 
in  Fig.  77,  constitute  what  is  called  an  astatic  system.  Such  a  sys- 
tem if  suspended  in  the  earth's  magnetic  field  would  point  indif- 
ferently in  any  direction  if  the  two  magnets  were  exactly  alike 
.and  exactly  opposite  in  direction.  If  the  two  needles  NS  and 
SN  are  nearly  alike  the  earth's  field  will  have  but  a  very  slight 
directing  action  upon  the  system.  Such  a  pair  of  magnetic 
needles  may  be  suspended  with  one  of  its  magnets  inside  of  a 
galvanometer  coil  as  shown  in  Fig.  78,  or  with  its  two  magnets 
surrounded  by  two  properly  connected  coils  as  shown  in  Fig.  79, 
and  the  result  will  be  an  extremely  sensitive  galvanometer.  The 
design  shown  in  Fig.  79  is  due  to  Lord  Kelvin.  A  galvanometer 
•constructed  after  this  design  with  very  short  magnetic  needles, 


MAGNETIC   EFFECT   OF    ELECTRIC   CURRENT. 


i  FIBRE 


!  FIBRE 


light  connecting  rod  and  mirror,  and  coils  containing  many  turns 
of  fine  wire  may  be  made  to  indicate  distinctly  a  current  as  small 
as  one  million-millionth  of  an  ampere  (io~12  ampere). 

The  Kelvin  galvanometer  may 
be  used  for  the  approximate  meas- 
urement of  very  weak  currents,  be- 
cause the  deflection,  within  a  small 
range,  is  proportional  to  the  cur- 
rent. 


COIL  I 


COIL  II 


Fig.  78. 


Fig.  79. 


61.    The  sensitive  galvanometer 

(U Arsonval    type).  —  A   coil  sus- 
pended in  a  magnetic  field  is  acted 
upon  by  a  torque  when  a  current 
flows  through  the  coil,  and  the  torque  is  given  by  the  equation 

(38),  namely: 

T=  abZIH 

as  explained  in  Art.  58.  If  the  coil  is  suspended  by  fine  wires 
this  torque  will  turn  the  coil  more  or  less,  and,  in  order  that  the 
coil  may  be  perceptibly  turned  by  a  very  weak 
current,  the  suspending  wires  (which  serve  to 
lead  current  to  and  from  the  coil)  must  be  very 
fine,  the  number  of  turns  of  wire  in  the  coil  must 
be  great,  and  the  magnetic  field  H  in  which  the 
coil  is  suspended  must  be  intense.  In  order  to 
obtain  a  quick  movement  of  the  coil  it  is  impor- 
tant to  have  its  breadth  b  moderately  small. 
Figure  80  shows  the  essential  parts  of  a  sen- 
sitive galvanometer  constructed  according  to 
these  principles.  It  consists  of  an  elongated 
coil  of  fine  wire  suspended  in  the  strong  field 
between  the  poles  of  a  magnet.  This  type  of 
galvanometer  is  due  to  D'Arsonval.  It  is  not 
so  sensitive  as  the  galvanometer  of  the  Kelvin 
type,  but  it  is  scarcely  affected  by  outside  magnetic  influences. 
9 


SUSPENDING 
WIRE 


O  MIRROR 


114        ELEMENTS  OF  ELECTRICITY  ^AND    MAGNETISM. 

The  D' Arson val  galvanometer  may  be  used  for  the  approx- 
imate measurement  of  weak  currents,  because  the  deflection, 
within  a  small  range,  is  proportional  to  the  current ;  that  is 

JJl 

i=kd 

in  which  /  is  the  current  flowing  through  the  galvanometer,  d 
is  the  deflection  in  scale  divisions,  and  k  is  a  proportionality 
factor  which  is  called  the  reduction  factor  of  the  galvanometer. 

PROBLEMS. 

80.  A  horizontal  wire  10  meters  long,  stretched  due  magnetic 
east  and  west,  is  pushed  up  by  the  horizontal  component  of  the 
earth's  field  with  a  force  of  2,500  dynes.     What  is  the  direction 
and  strength  of  the  current  in  the  wire  ?     The  horizontal  com- 
ponent of  the  earth's  field  is  0.2  gauss.     Ans.  125  amperes  east. 

81.  The  armature  of  a  dynamo  has  a  length,  under  the  pole- 
face,  of  30  cm.     The  magnetic  field  intensity  between  the  pole- 
face  and  the  armature  core  is  6,000  gausses.     The  surface  of  the 
armature  is  covered  with  straight  wires  parallel  to  the  axis  of  the 
armature.      Each  of  these  wires  carries  a  current  of  75  amperes. 
Calculate  the  force  acting  on  each  wire.     Ans.   1,350,000  dynes. 

82.  A  horizontal  electric  light  wire   stretched   due   magnetic 
north  and  south  carries  1,000  amperes  of  current  flowing  towards 
the  north.     The  length  of  the  wire  is  250  meters,  the  intensity 
of  the  earth's  field  is  0.57  gauss  and  the  magnetic  dip  is  63°. 
Find  the  value  of  the  force  pushing  on  the  wire  and  specify  its 
direction.     Ans.    1,269,500  dynes  west. 

83.  A  circular  coil  of  wire  of  20  cm.  radius  has  15  turns  of 
wire.      How  much  current  is  required  in  the  coil  to  produce  at 
the  center  of  the  coil  a  field  intensity  of  o.  57  gauss  ?     Ans.  o.  1 2 1 
abamperes. 

84.  The  two  straight  parallel   wires  of  an  electric  light  pole-line  are  35  inches 
apart  center  to  center,  and  a  current  of  500  amperes  is  flowing  out  in  one  wire  and 
back  in  the  other.      Find:   (a)  The  intensity  of  the  magnetic  field  due  to  the  wires  at 
a  point  midway  between  them,  and  (b]  the  intensity  of  the  magnetic  field  due  to  the 


MAGNETIC    EFFECT   OF   ELECTRIC   CURRENT.         115 

two  wires  at  a  point  which  is  21  inches  from  the  axis  of  one  wire  and  28  inches  from 
the,. axis  of  the  other  wire.     Ans.    (a]  4.5  gausses,      (b)  2.34  gausses. 

85.  A  thin  brass  tube  2  inches  in  diameter  and  6  feet  long  is 
wound  with  1,400  turns  of  wire,     (a)  Calculate  the  field  intensity 
inside  of  this  coil  when  a  current  of  5  amperes  flows  through  the 
wire.     (&)  Calculate  the  total  energy  of  the  magnetic  field  inside 
of  the  coil.     Ans.  (a)  48.1  gausses,     (b)  341,220  ergs. 

86.  A  tangent  galvanometer  gives  a  deflection  of  10°  for  1.2 
amperes.     Calculate  the  deflection  which  will  be  produced  by  1 5 
amperes.     Ans.  65°  35'. 

87.  A  rectangular  frame  25  x  40  cm.  has   10  turns  of  wire 
wound  upon  it.     The  frame  is  balanced  horizontally  upon  an  axis 
pointing  due  magnetic  east  and  west.     A  current  of  28  amperes 
is  sent  through  the  wire.      Required  the  distance  from  the  axis  at 
which  a  lo-gram  (9,8oo-dyne)  weight  must  be  hung  to  balance 
the  torque  action  due  to  the  earth's  magnetic  field  at  a  place 
where  its    intensity  is    0.57    gauss  and  its   dip  is  63°.      Ans. 
0.74  cm. 

88.  A  circular  coil  has   100  turns  of  wire.     The  diameter  of 
the  mean  turn  is   1 6  centimeters,  and  a  current  of  1 5   amperes 
flows  through  the  coil.     This  coil  is  suspended  with  its  plane 
lying  vertical  and  magnetic  north  and  south,     (a)  Calculate  the 
torque  in  dyne-centimeters  with  which  the  horizontal  component 
of  the  earth's  field  (0.2)  acts  upon  the  coil  and  specify  the  direc- 
tion of  the  axis  about  which  this  torque  is  exerted,     (b)  Calcu- 
late the  torque  in  dyne  centimeters  with  which  the  vertical  com- 
ponent of  the  earth's  field  (0.68)  acts  on  the  coil  and  specify  the 
direction  of  the  axis  about  which  this  torque  is  exerted.     Ans.  (a) 
Axis,  vertical  ;  torque,  6,032  dyne-centimeters,      (b)  Axis,  north 
and  south  ;  torque,  20,508  dyne-centimeters. 

89.  A  circular  coil  10  cm.  in  diameter,  having  50  turns  of  wire,  is  hung  by  a  phos- 
phor-bronze wire  at  the  center  of  a  large  circular  coil   120  cm.  in  diameter,  having 
500  turns  of  wire.     The  suspending  wire  is  free  from  twist  when  the  planes  of  the 
two  coils  are  at  right  angles,  and  a  torque  of  250  dyne-centimeters  twists  the  wire 
through  one  radian  of  angle.      How  much  current  must  pass  through  the  two  coils  in 
series  to  cause  the  suspended  coil  to  turn  30°  from  its  position  of  equilibrium  ?    What 


Il6        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

happens  if  the  current  is  reversed  in  one  coil  ?     What  happens  if  the  current  is  re- 
versed in  both  coils?     Ans.  0.27  ampere. 

90.  The   spiral   spring   of  a    Seimens   electrodynamometer  is 
twisted  through  an  angle  of  225  °  to  balance  the  force  action  on 
the  movable  coil  when  a  current  of  14  amperes  flows  through 
the  instrument.     A  twist  of  1 60°  is  required  to  balance  the  force 
action  of  a  current  which  is  being  measured  by  the  instrument. 
Required  the  value  of  this  current.     Ans.    n.8  amperes. 

91.  The  horizontal  component  of  the  earth's  magnetic  field  at 
the  needle  of  a  sensitive  galvanometer  (Kelvin  type)  is  o.  18  gauss, 
and  its  direction  is  due  north.     It  is  desired  to  produce  at  the 
needle  a  resultant  magnetic  field  of  0.02  gauss  intensity  in  a  due 
easterly  direction.     Find  the  distance  and  direction  from  the  gal- 
vanometer needle  at  which  an  isolated  north  magnet  pole   of 
strength  600  gausses  must  be  placed  to  produce  the  desired  re- 
sult.    Ans.   57.6  cm.,  6°  20'  west  of  north. 


CHAPTER  V. 


INDUCED   ELECTROMOTIVE   FORCE. 

THE  DYNAMO. 
62.  Lenz's  law.    Electromagnetic  theory  a  branch  of  mechanics.* 

—  The  idea  of  electric  current  is  strictly  analogous  to  the  mechan- 
ical idea  of  velocity  f  and  an  insight  into  the  nature  of  induced 
electromotive  force  can  be  obtained  only  by  drawing  a  parallel 
between  the  equations  in  Mechanics  and  the  equations  of  Elec- 
tricity and  Magnetism. 

The  product  of  the  force  F  exerted  on 
a  body  which  moves  at  velocity  v  in  the 
direction  of  F  is  equal  to  the  power  P 
developed  by  the  agent  which  is  exerting 
the  force  on  the  body  ;  that  is 

P=Fv 


The  product  of  the  electromotive  force 
E  of  a  generator  and  the  current  7  de- 
livered by  the  generator  is  equal  to  the 
power  P  delivered  by  the  generator  to 
the  circuit  to  which  the  generator  delivers 
current.  That  is, 


in  which  P  is  expressed  in  ergs  per  sec- 
ond if  E  is  expressed  in  abvolts  and  1 
in  abamperes,  or  P  is  expressed  in  watts 
if  E  is  expressed  in  volts  and  /  in 
amperes. 

In  order  to  produce  a  current  I  through 
a  circuit  of  which  the  resistance  is  R,  an 
electromotive  force  equal  to  RI  is  re- 
quired ;  that  is, 

E  =  RI 

Multiplying  both  members  of  this  equation 


in  which  P  is  expressed  in  ergs  per  sec- 
ond if  F  is  expressed  in  dynes  and  v  in 
centimeters  per  second.  There  are  no 
names  for  the  units  of  force  and  velocity 
which  correspond  to  the  watt  as  a  unit 
of  power. 

A  force  F  acts  upon  a  boat  and  in- 
creases the  velocity  v  of  the  boat  until 
all  of  the  force  F  is  used  to  overcome  the 
friction  of  the  water.  Let  us  assume  that 
the  friction  of  the  water  is  proportional 
to  the  velocity  of  the  boat,  or  equal  to  rv 


*  The  mechanical  analogies  which  are  outlined  in  this  article  are  exact  and  com- 
plete. Any  one  who  is  interested  in  the  full  details  of  this  matter  should  read  a  re- 
markable paper  on  The  Motion  of  Monocyclic  Systems  by  H.  von  Helmholtz,  Crelle's 
Journal,  Vol.  97,  pp.  in  and  317.  A  very  interesting  and  instructive  book  entitled 
Applications  of  Dynamics  to  Physics  and  Chemistry,  by  J.  J.  Thomson,  touches  in- 
directly upon  this  matter.  See  also  Art.  125  of  this  text. 

t  Electric  current  is  velocity  and  it  is  entirely  meaningless  to  speak  of  the 
velocity  with  which  an  electric  current  flows  along  a  wire.  This  matter  will  be  made 
clear  when  we  come  to  discuss  electric  waves. 

117 


Il8        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 


by  the  current  and  remembering  that  El 
is  equal  to  the  power  delivered  to  the  cir- 
cuit, we  have 

P=RI* 

In  these  equations  R  may  be  expressed 
in  ohms,  /  in  amperes,  E  in  volts  and 
P  in  watts,  or  R  may  be  expressed  in 
abohms,  /  in  abamperes,  E  in  abvolts 
and  P  in  ergs  per  second. 


where  r  is  a  constant.     Then  we  have 
F=rv 

Multiplying  both  members  of  this  equa- 
tion by  v  and  remembering  that  Fv  is 
the  power  that  is  delivered  to  the  boat, 
we  have 


In  these  equations  c.g.s.  units  are  most 
conveniently  used,  that  is,  F  is  expressed 
in  dynes,  v  in  centimeters  per  second, 
and  P  in  ergs  per  second.  The  coefficient 
r  is  exactly  analogous  to  the  resistance 
of  an  electric  circuit. 

Consider  the  wires  on  the  surface  of  the  cylinder  A  in  Fig.  63 
with  electric  currents  flowing  through  them  as  indicated  by  the 
dots  and  crosses.  These  wires  are  pushed  sidewise  by  the  mag- 
netic field,  as  explained  in  Art.  51,  and,  if  the  cylinder  A  is 
allowed  to  turn  so  that  the  wires  A  move  with  *  this  side  force, 
mechanical  work  is  obtained.  Where  does  this  work  come  from  ? 
If  the  cylinder  A  is  forcibly  turned  so  that  the  wires  move  against 
the  side  force  with  which  the  magnetic  field  pushes  on  them, 
mechanical  work  is  expended.  Where  does  this  work  go  to  ? 
The  present  chapter  is  devoted  to  the  consideration  of  these  two 
questions,  and  some  idea  of  the  conclusions  which  will  be  reached 
may  be  obtained  by  a  brief  discussion  of  the  analogous  mechanical 
problem.  A  person  standing  on  the  swinging  span  of  a  draw- 
bridge as  shown  in  Fig.  8  1  is  acted  upon  by  a  centrifugal  force,  as 
indicated  by  the  arrow,  and  this  centrifugal  force  depends  upon  the 
angular  velocity  of  the  moving  span.  If,  while  the  span  is  swing- 
ing, the  person  walks  towards  the  center  of  the  span,  he  does  work 
in  moving  himself  against  the  centrifugal  force  ',  and  this  work  helps 
to  turn  the  span.  If  the  person  walks  away  from  the  center  of  the 
swinging  span  he  is  helped  by  the  centrifugal  force,  or,  in  other 
words,  he  receives  energy  or  work  from  the  swinging  span,  more 

*  A  body  is  said  to  move  with  a  force  which  acts  upon  it  when  it  moves  in  the 
direction  of  the  force.  A  body  is  said  to  move  against  a  force  which  acts  upon  it 
when  it  moves  in  a  direction  opposite  to  the  direction  of  the  force. 


INDUCED  ELECTROMOTIVE  FORCE. 


119 


work  is  required  to  keep  the  span  turning  than  would  otherwise  be 
necessary,  and  the  work  received  by  the  moving  person  is  equal  to 
the  additional  work  so  expended  in  turning  the  span. 


\axis  of  rotation 


Fig.  81. 

Inasmuch  as  the  idea  of  electric  current  strength  is  strictly 
analogous  to  the  mechanical  idea  of  velocity,  the  question  as  to 
what  becomes  of  the  work  done  in  moving  a  wire  against  a  force 
which  depends  on  the  current  is  strictly  analogous  to  the  question 
as  to  what  becomes  of  the  work  done  in  moving  a  body  against  a 
force  which  depends  on  velocity.  Therefore  the  above  example  of 
a  person  moving  radially  on  a  swinging  bridge  span  is  analogous 
to  the  following :  A  wire  is  connected  to  a  battery  so  that  an 
electric  current  flows  through  it,  and  the  wire  is  stretched  across 
a  magnetic  field  as  shown  in  Fig.  6 1 .  Under  these  conditions  the 
magnetic  field  pushes  sidewise  on  the  wire,  and  this  side  force 
depends  on  the  current.  If  the  wire  be  moved  sidewise  against 
this  force,  work  has  to  be  done  and  this  work  helps  to  maintain  the 
current.  If  the  wire  is  moved  in  the  direction  of  the  side  force r,  the 
side  force  does  work  in  helping  to  move  the  wire,  more  work  is  re- 
quired to  keep  the  current  flowing  than  would  otherwise  be  neces- 
sary, and*  the  work  received  by  the  moving  wire  is  equal  to  the  ad- 
ditional work  thus  done  in  keeping  the  current  flowing. 

In  the  example  of  the  swinging  bridge  span,  the  force  exerted 
l>y  the  engine  which  drives  the  span  must  be  supposed  to  be 
greater  or  less  according  as  the  man  is  moving  outwards  or  in- 
wards (with  or  against  the  centrifugal  force)  if  the  velocity  of 
turning  is  to  be  kept  constant.  In  the  example  of  the  moving 


120        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

wire,  the  battery  which  supplies  the  electric  current  must  be  sup- 
posed to  have  a  greater  or  less  electromotive  force  according  as 
the  wire  is  moving  with  or  against  the  side  force  due  to  the  mag- 
netic field  if  the  strength  of  the  -Current  is  to  be  kept  constant. 

The  action  described  above  in  connection  with  the  motion  of  a 
man  on  a  swinging  bridge  span  may  be  perceived  in  a  very  strik- 
ing way  by  holding  weights  in  one's  hands,  swinging  round  and 
round  on  one's  heel,  and  drawing  the  weights  inwards  or  extend- 
ing them  outwards  repeatedly. 

The  facts  outlined  above  in  connection  with  the  moving  wire, 
constitute  what  is  called  Lenzs  law,  a  more  elaborate  statement 
of  which  will  be  given  later. 

63.  Induced  electromotive  force.  —  Faraday  discovered  in  1831 
that  a  momentary  electric  current  is  produced  in  a  coil  of  wire 
when  a  magnet  is  thrust  into  a  coil  or  withdrawn  from  the  coil, 
or  when  an  iron  rod  upon  which  the  coil  is  wound  is  magnetized 
or  demagnetized.  The  motion  of  the  magnet  in  the  first  case  or 
the  varying  magnetism  of  the  iron  rod  in  the  second  case,  pro- 
duces a  momentary  electromotive  force  in  *  the  coil  and  this  elec- 
tromotive force  in  its  turn  produces  a  momentary  current  if  the 
coil  forms  a  portion  of  a  closed  circuit.  The  electromotive  force 
and  electric  current  produced  in  this  way  are  called  induced  elec- 
tromotive force  and  induced  current. 

Examples  of  Lenz's  law.  —  A  current  induced  in  a  coil  when  a 
magnet  is  thrust  into  the  coil  is  in  such  a  direction  as  to  tend  to 
push  the  magnet  out  of  the  coil,  and  the  work  done  in  moving  the 
magnet  against  this  opposing  force  is  the  work  which  goes  to 
produce  the  induced  current.  The  current  induced  in  a  coil  when 
a  magnet  is  withdrawn  from  the  coil  is  in  such  direction  as  to 
tend  to  draw  the  magnet  into  the  coil  and  the  work  done  in  mov- 
ing the  magnet  against  this  opposing  force  is  the  work  which 
goes  to  produce  the  induced  current.  When  an  iron  rod  with  a 

*One  should  always  speak  of  the  electromotive  force  between  two  points,  never  of 
the  electromotive  force  in  a  circuit,  except  only  when  one  is  speaking  of  an  induced 
electromotive  force. 


INDUCED  ELECTROMOTIVE  FORCE. 


121 


short-circuited  winding  of  wire  is  magnetized,  the  current  induced 
in  the  winding  opposes  the  magnetization  and  more  work  is  re- 
quired to  magnetize  the  rod  than  would  be  required  if  the  in- 
duced current  did  not  exist.  This  additional  work  is  that  which 
produces  the  induced  current. 

64.  Electromotive  force  induced  in  a  straight  wire  moving  side- 
wise  across  a  uniform  magnetic  field.  —  Consider  a  straight  wire 
BB' ',  Fig.  82,  which  slides  sidewise  at  a  velocity  of  v  centime- 

B 


Fig.  82. 

ters  per  second  along  two  straight  wires  or  rails  AB  and  A'Br , 
distant  /  centimeters  from  each  other.  The  rails  AB  and 
AfB'  are  connected  at  AA'  so  that  ABB' A1  is  a  closed  cir- 
cuit. The  whole  arrangement  is  placed  in  a  uniform  magnetic 
field  of  intensity  H,  the  direction  of  the  field  being  perpendic- 
ular to  the  plane  of  the  figure  and  towards  the  reader.  The 
motion  of  the  wire  BB'  induces  in  it  an  electromotive  force  E 
which  in  its  turn  produces  a  current  /  in  the  circuit  ABB'A', 
and  because  of  this  current  the  magnetic  field  pushes  the  wire 
BB1  sidewise  with  a  force  F  as  indicated  in  the  figure.  The 
rate  at  which  work  is  done  in  moving  the  wire  BB'  against  the 
force  F  at  velocity  v  is  Fv  ergs  per  second,  and  the  rate  at 
which  work  is  done  by  the  electromotive  force  E  in  maintaining 
the  current  /  is  El  ergs  per  second,  E  being  expressed  in 
abvolts,  and  7  being  expressed  in  abamperes.  According  to 
Lenz's  law,  the  work  done  in  moving  the  wire  BB'  against  the 
force  F  goes  to  maintain  the  current.  Therefore  we  have 


122        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

Fv  =  El 
and  from  equation  (28),  in  Art.  52, 


whence,  substituting  this  value  of  F  in  equation  (i),  we  have 

E=lHv  «.  (42) 

that  is,  the  electromotive  force  induced  in  a  wire  /  centimeters 
long,  moving  sidewise  at  a  velocity  of  v  centimeters  per  second 
across  a  uniform  magnetic  field  of  intensity  H  is  equal  to  the 
product  IHv.  This  product  expresses  the  induced  electromotive 
force  in  c.g.s.  units  or  ab  volts,  one  abvolt  being  an  electromotive 
force  which  will  do  work  at  the  rate  of  one  erg  per  second 
in  maintaining  a  current  of  one  abampere.  One  volt  equals 
i  o8  abvolts. 

65.  Expression  of  induced  electromotive  force  in  terms  of  lines 
of  force  cut  per  second.  —  -During  t  seconds  the  sliding  piece 
BB*  ',  Fig.  82,  moves  over  a  distance  vt  and  sweeps  over  Ivt 
square  centimeters  of  area.  The  product  of  this  area  by  the  field 
intensity  H  gives  the  number  of  lines  of  force  <I>  which  pass 
through  the  area  according  to  equation  (18),  and  this  is  the 
number  of  lines  of  force  cut  by  the  moving  wire  in  t  seconds, 

that  is, 

3>  =  IHvt  (i) 

Dividing  both  members  of  this  equation  by   /,    we  have 


but  &Jt  is  the  rate  at  which  the  moving  wire  BB'  cuts  lines 
of  force,  or,  in  other  words,  it  is  the  number  of  lines  of  force  cut 
per  second,  and  IHv  is  the  electromotive  force  in  abvolts  induced 
in  the  wire,  according  to  equation  (42).  Therefore  the  electro- 
motive force  in  abvolts  induced  in  a  moving  wire  is  equal  to  the 
number  of  lines  of  force  cut  per  second  by  the  moving  wire.  This 
result  is  true  for  any  wire,  straight  or  curved,  moving  in  any 


INDUCED   ELECTROMOTIVE   FORCE.  123 

manner  in  any  magnetic  field,  uniform  or  non-uniform,  although 
the  derivation  here  given  applies  to  the  motion  of  a  straight  wire 
across  a  uniform  field. 

66.  Expression  of  induced  electromotive  force  in  terms  of  rate  of 
change  of  magnetic  flux  through  a  circuit.* — The  total  magnetic 
flux  through  the  circuit  ABB'  A'  ^  Fig.  82,  is  given  by  equation 
(i),  Art.  65,  and  the  rate  at  which  the  moving  wire  BBr  cuts 
lines  of  force  is  the  rate  of  increase  of  <I>.  Therefore  the  electro- 
motive force  induced  in  a  circuit  is  equal  to  the  rate  of  change  of 
the  magnetic  flux  through  the  circuit,  that  is, 

(43) 

Experiment  shows  this  equation  to  be  true  when  the  change  of 
magnetic  flux  is  due  to  motion  and  also  when  the  change  of  mag- 
netic flux  is  due  to  varying  strength  of  the  magnetic  field. 

The  negative  sign  in  equation  (43)  has  no  immediate  importance. 
It  is  chosen  in  accordance  with  the  following  convention.  A 
right  handed  screw  with  its  axis  parallel  to  the  magnetic  field  H 
(directed  towards  the  reader  in  Fig.  82)  would  have  to  be  turned 
in  a  direction  opposite  to  the  flow  of  induced  current  produced  by 
an  increasing  flux  in  order  to  make  the  screw  travel  in  the  direc- 
tion of  H.  It  is  therefore  convenient  to  look  upon  the  induced 
current  or  the  induced  electromotive  force  as  negative  when 
d^ldt  is  positive. 

Equation  (43)  expresses  the  electromotive  force  induced  in  a 
single  turn  of  wire.  When  a  region  of  changing  magnetic  flux  is 
surrounded  by  Z  turns  of  wire,  then  equation  (43)  expresses  the 
electromotive  force  induced  in  each  turn  of  wire,  and  the  total 
electromotive  force  is 

*— •*£  (44) 

*  Let  it  be  remembered  that  the  fundamental  action  upon  which  induced  electro- 
motive force  depends  is  the  cutting  the  lines  of  force  by  a  moving  conductor  or  the 
sweeping  of  moving  lines  of  force  past  a  stationary  conductor. 


124        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

67.  The  dynamo. — The  dynamo  is  a  machine  for  the  produc- 
tion and  maintenance  of  an  electric  current  when  the  machine  is 
supplied  with  mechanical  power,  or,  conversely,  for  the  develop- 
ment of  mechanical  power  whe"h  the  machine  is  supplied  with 
electric  current.  When  used  for  the  former  purpose  the  dynamo 
is  called  an  electric  generator,  and  when  used  for  the  latter  pur- 
pose, the  dynamo  is  called  an  electric  motor. 

The  action  of  the  dynamo  as  a  generator  is  essentially  as 
follows :  A  wire  is  forced  by  an  external  source  of  mechanical 
power  to  move  sidewise  across  a  magnetic  field.  This  motion 
induces  an  electromotive  force  in  the  wire  and  this  electromotive 
force  produces  a  current  in  the  circuit  which  is  connected  to  the 
ends  of  the  wire.  The  induced  current  causes  the  magnetic  field 
to  push  on  the  moving  wire  in  a  direction  opposite  to  its  motion, 
and  the  work  done  in  overcoming  this  opposing  force  is  the  work 
that  goes  to  maintain  the  induced  current. 

The  action  of  the  dynamo  as  a  motor  is  essentially  as  follows: 
An  electric  current  from  an  external  source  is  forced  through  a 
wire  which  is  allowed  to  move  sidewise  in  a  magnetic  field  in  the 
direction  of  the  side  push  upon  it,  thus  developing  mechanical 
power.  The  motion  of  the  wire  induces  in  it  an  electromotive 
force  which  opposes  the  flow  of  current  through  the  wire,  and  the 
work  done  by  the  external  source  of  electric  current  in  forcing 
the  current  through  the  wire  in  opposition  to  this  induced  electro- 
motive force  is  the  work  which  appears  as  mechanical  energy  in 
the  motor. 

The  above-described  action  of  the  dynamo  as  a  generator  and 
as  a  motor  constitutes  a  complete  statement  of  what  is  called 
Lenz's  Law,  namely,  that  an  induced  current  leads  to  the  pro- 
duction of  a  force  which  opposes  the  action  which  produces  it 
and  the  work  done  in  overcoming  this  opposing  force  is  the  work 
that  goes  to  produce  the  induced  current. 

Types  of  dynamos. — There  are  two  distinct  types  of  dynamo 
electric  machines,  namely,  (a)  alternating-current  machines  and 
(&)  direct-current  machines.  The  alternating-current  generator 


INDUCED  ELECTROMOTIVE  FORCE. 


125 


delivers  what  is  called  an  alternating  current,  that  is,  a  current 
which  is  subject  to  rapid  periodic  reversals  of  direction.  The 
direct-current  generator,  on  the  other  hand,  delivers  a  current 
which  is  not  reversed  in  direction  and  which  is  usually  quite 
steady  in  value. 

68.  The  alternating-current  dynamo.  —  The  simplest  form  of 
the  alternating-current  dynamo  is  shown  in  Fig.  83.     A  wire    Wy 


Fig.  83. 

perpendicular  to  the  plane  of  the  paper,  is  moved  sidewise  along 
the  dotted  line  so  as  to  cut  the  magnetic  lines  of  force  which 
emanate  from  the  inwardly  projecting  poles  NSNS  of  a  large 
electromagnet  which  is  called  the  field  magnet  of  the  alternator. 
While  the  wire  is  sweeping  across  a  north  pole  an  electromotive 
force  is  induced  in  it  in  one  direction,  and  while  the  wire  is  sweep- 
ing across  a  south  pole  an  electromotive  force  is  induced  in  it  in 
the  opposite  direction.  This  repeatedly  reversed  electromotive 
force  is  called  an  alternating  electromotive  force  and  it  produces  an 
alternating  current  in  the  wire  and  in  an  outside  circuit  to  which 
the  ends  of  a  wire  may  be  connected. 

In  commercial  alternators  large  numbers  of  wires  are  used  in- 
stead of  the  single  wire  W  shown  in  Fig.  83,  and  these  wires 
are  placed  in  slots  in  the  periphery  of  a  rotating  cylindrical  mass 
of  laminated  iron.  Thus,  Fig.  84  shows  4  wires  in  4  slots  and 


126        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 


Fig.  85  shows  1 6  wires  in  16  slots.  Figures  84^  and  85^  are 
what  are  called  developed  diagrams  which  show  how  the  wires 
are  connected  to  each  other  and  how  they  are  connected  to  two 


To  receiving  circuit 
•\  -* 


r" 

— 

— 

-i 

"  "J       1 

j  t 

1 

• 

!   1 

, 

!     ! 

i 

, 

! 

L  N 

s  :   ;  N  i 

U,_j  1           1  +.  1 

.JLJ 

I J 


Fig.  84a- 


Fig.  84b. 


J 

To  receiving  circuit 

Lc 

*^  J 

r 

j 

x  — 

2 

N 

il 

1       1 

J  I. 

'1   r" 
| 

,  L 

'(—VI1 

"i  1 

,       js 
!     ! 

..J         1  

1 

-  J 

Fig.  85a. 


Fig.  85b. 


insulated  metal  rings  r  and  rf  upon  which  two  metal  brushes 
a  and  b  rub,  thus  keeping  the  moving  wires  connected  to  an 
outside  receiving  circuit. 

The  laminated  iron  cylinder  A  A  with  its  winding  of  wire  is 
called  the  armature  of  the  alternator,  the  metal  rings  r  and  r' 
are  called  collector  rings.  The  field  magnet  of  an  alternator  must 


INDUCED  ELECTROMOTIVE  FORCE. 


127 


be  excited  by  direct  current  which  is  generally  supplied  by  a 
small  auxiliary  direct-current  generator  called  the  exciter. 

Definition  of  the  cycle.  Frequency.  —  The  electromotive  force 
of  an  alternator  passes  through  a  set  of  positive  values  while  a 
group  of  armature  wires  is  passing  a  north  pole  of  the  field 
magnet,  and  through  a  set  of  negative  values  while  the  given 
group  of  armature  wires  is  passing  a  south  pole  of  the  field  mag- 
net. The  complete  set  of  values,  including  positive  and  negative 
values,  is  called  a  cycle,  the  duration  of  a  cycle  is  called  a  period, 
and  the  number  of  cycles  per  second  is  called  tite  frequency.  If 
the  field  magnet  of  an  alternator  has  /  poles  (pJ2  north  poles 
and  p/2  south  poles),  then  the  frequency  of  its  electromotive 
force  is  pnJ2,  where  n  is  the  speed' of  the  alternator  armature 
in  revolutions  per  second.  This  is  evident  when  we  consider 
that  a  complete  cycle  corresponds  to  the  passage  of  a  given  group 
of  armature  wires  across  two  field  poles,  a  north  pole  and  a  south 
pole,  so  that  there  are  pJ2  cycles  in  one  revolution.  The 

standard  frequencies  of  com- 
mercial alternators  in  practice 
are  25  cycles  per  second  for 
large  installations  for  the 
transmission  of  power,  60  cy- 
cles per  second  for  alternators 
which  supply  current  for  both 
lamps  and  motors,  and  133 
cycles  per  second  for  the  older 
styles  of  alternators  which 
supply  current  to  lamps  only. 


69.  The  direct-current  dy- 
namo is  somewhat  more  com- 
plicated than  the  alternator. 
The  following  description  ap- 
plies to  the  direct-current  dynamo  having  an  armature  of  the  so- 
called  ring  type,  and  having  a  bipolar  field  magnet.  An  iron  ring 


Fig.  86. 


128        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 


ab  which  is  built  up  of  sheet  iron  stampings,  is  wound  uniformly 
with  insulated  wire  as  indicated  in  Fig.  86,  the  ends  of  the  wire 
being  spliced  together  and  soldered  so  that  the  winding  is  endless. 
This  iron  ring  with  its  windingrbf  wire  is  called  the  armature  of 
the  machine,  and  it  rotates,  as  indicated  by  the  curved  arrow,  be- 
tween the  poles  of  a  strong  field  magnet. 

The  wires  on  the  outside  of  the  iron  ring 'have  electromotive 
forces  induced  in  them  as  they  move  across  the  pole  faces  of  the 
field  magnet  and  cut  the  lines  of  force.  These  electromotive 
forces  cannot,  however,  produce  current  in  the  endless  wire  that 
is  wound  on  the  armature,  because  exactly  equal  and  opposite 
electromotive  forces  are  induced  on  the  opposite  sides  c  and  d 
of  the  ring,  as  shown  diagrammatically  in  Fig.  87  in  which  the 


Fig.  87. 


Fig.  88. 


circle  adbc  represents  the  endless  wire  on  the  ring.  A  steady, 
or  very  nearly  steady,  current  can,  however,  be  taken  from  the 
winding  on  the  ring  by  keeping  the  terminals  of  an  external  cir- 
cuit /,  Fig.  88,  in  metallic  contact  with  the  windings  on  the  ring 
at  a  and  b.  For  this  purpose  the  insulation  may  be  removed 
from  the  outer  portions  of  the  wire  windings  on  the  ring  and  two 
stationary  metal  or  carbon  brushes  SS,  Fig.  88,  may  be  ar- 
ranged to  rub  at  a  and  b  as  the  ring  rotates.  In  practice  wire 
leads  are  soldered  to  the  various  turns  of  wire  on  the  ring  and 
connected  to  insulated  copper  bars  near  the  axis  of  rotation  as 
shown  in  Fig.  89.  Sliding  contact  is  then  made  with  these 
copper  bars  instead  of  with  the  turns  of  wire  at  a  and  b 


INDUCED  ELECTROMOTIVE  FORCE. 


I29 


directly.     This  set  of  copper  bars  constitutes  what  is  called  the 
commutator. 

Shunt  and  series  field  windings.  —  The  field  magnet  of  a  direct- 
current  generator  is  usually  excited  by  current  taken  from  the 
machine  itself.  The  winding  of  wire  on  the  field  magnet  may 
consist  of  many  turns  of  comparatively  fine  wire  having  a  con- 


Fig.  89. 

siderable  resistance.  In  this  case  the  terminals  of  the  field  wind- 
ing are  connected  directly  to  the  brushes  of  the  machine,  and 
from  two  to  ten  per  cent,  of  the  permissible  current  output  of  the 
generator  flows  through  the  field  windings  and  excites  the  field, 
the  remainder  of  the  permissible  current  output  being  available 
for  use  in  the  external  circuit.  In  this  case  the  field  winding  and 
the  outside  receiving  circuit  are  in  parallel  with  each  other  between 
the  brushes,  so  that  the  field  winding  is  in  the  relation  of  a  shunt 
to  the  outside  receiving  circuit.  A  direct-current  dynamo  with 
its  field  windings  arranged  in  this  way  is  called  a  shunt  dynamo. 

The  winding  of  wire  on  the  field  magnet  of  a  direct-current 
dynamo  may  consist  of  comparatively  few  turns  of  heavy  wire 
having  a  low  resistance.  In  this  case  the  field  winding  is  con- 
nected in  series  with  the  external  receiving  circuit,  the  whole  cur- 
rent delivered  by  the  machine  flows  through  the  field  winding, 
and  from  two  to  ten  per  cent,  of  the  electromotive  force  developed 
by  the  machine  is  used  to  force  the  current  through  the  field 
winding,  the  remainder  being  available  for  forcing  current  through 
the  external  receiving  circuit.  A  direct-current  dynamo  with  its 
field  windings  arranged  in  this  way  is  called  a  series  dynamo. 

10 


130        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

The  multipolar  direct-current  dynamo.  —  Figure  90  shows  a 
ring  armature  rotating  inside  of  a  crown  of  six  inwardly  project- 
ing field  magnet  poles.  The  electromotive  force  induced  in  the 
windings  as  they  sweep  across  the  pole  faces  cannot  produce  cur- 
rent in  the  endless  wire  that  is  wound  on  the  ring,  because  the 
electromotive  forces  induced  under  the  north  poles  are  just 
balanced  by  the  electromotive  forces  induced  under  the  south 
poles,  as  shown  diagrammatically  in  Fig.  91.  To  utilize  the  in- 


Fig.  90. 


Fig.  91 


duced  electromotive  forces  eeeeee,  Fig.  91,  for  the  production  of 
direct  current,  six  brushes  aaa  and  bbb,  Fig.  90,  should  be  used. 
Three  of  these  brushes  maintain  contact  with  the  windings  at 
aaa,  and  through  all  three  of  these  brushes  current  flows  out  of 
the  armature  to  one  terminal  of  a  receiving  circuit.  The  other 
three  brushes  maintain  contact  with  the  windings  at  bbb,  Fig.  91, 
all  three  of  these  brushes  are  connected  to  the  other  terminal  of 
the  receiving  circuit  and  current  flows  into  the  armature  through 
all  three.  The  three  brushes  aaa  together  constitute  the  positive 
terminal  of  the  armature,  and  the  three  brushes  bbb  together 
constitute  the  negative  terminal  of  the  armature. 

Number  of  current  paths  in  the  armature  between  positive  and 
negative  brushes.  —  In  the  bipolar  direct-current  dynamo  two 
brushes  are  used  as  shown  in  Fig.  88,  and  the  current  which 
enters  the  armature  at  the  negative  brush  divides  into  two  parts 


10 


INDUCED   ELECTROMOTIVE   FORCE.  13! 

and  flows  through  two  distinct  paths  in  the  armature  winding  to 
reach  the  positive  brush.  In  the  particular  multipolar  dynamo 
shown  in  Figs.  90  and  91  the  current  enters  the  armature  through 
three  brushes,  and  the  current  which  enters  at  each  of  the  three 
brushes  divides  into  two  parts  and  flows  through  two  distinct  paths 
to  reach  a  positive  brush.  Therefore  in  this  particular  machine, 
having  six  field  poles,  there  are  six  current  paths  through  the 
armature  from  negative  to  positive  brushes.* 

70.  Fundamental  equation  of  the  direct-current  dynamo.  —  Let  $  be  the  mag- 
netic flux  which  enters  the  armature  from  the  north  pole  of  the  field  magnet  and  leaves 
the  armature  at  the  south  pole  of  the  field  magnet,  let  Z  be  the  number  of  conductors 
on  the  outside  surface  of  the  armature,  let  n  '  be  the  speed  of  the  armature  in  revolu- 
tions per  second,  and  let  Ea  be  the  electromotive  force  induced  in  the  armature  wind- 
ing. A  voltmeter  connected  to  the  brushes  of  the  dynamo  would  indicate  the  value 
of  Eg,  if  the  current  in  the  armature  were  negligibly  small  ;  when  the  current  in  the 
armature  is  large,  a  portion  of  Ea  is  used  to  overcome  the  armature  resistance.  The 
equation  which  expresses  the  relation  between  Ea,  4>,  Z,  and  n  is  called  the  funda- 
mental equation  of  the  dynamo.  This  equation  is  here  derived  for  the  simplest  case, 
namely,  that  of  a  bipolar  dynamo  with  simple  ring-wound  armature.  In  this  case 

Ea  —  $Zn  abvolts 


2?.  =~    volts  (45*) 

Proof  of  equation  (45#).  —  During  i/n  second  the  armature  makes  one  complete 
revolution,  so  that  during  i/2n  second  a  given  conductor  sweeps  past  a  field  pole 
from  a  to  b  in  Fig.  86  and  cuts  $  lines  of  force.  Therefore  this  conductor  cuts 
lines  of  force  at  an  average  rate  which  is  equal  to  $-^  i/2«,  or  2$n  lines  of  force 
per  second  ;  which  is  equal  to  the  average  electromotive  force  induced  in  the  given 
conductor  during  the  time  that  it  is  moving  from  a  to  b  in  Fig.  86  ;  also  this  is  the 
average  electromotive  force  in  all  of  the  conductors  between  a  and  b  at  any  instant. 
Therefore,  since  there  are  Z/2  armature  conductors  or  wires  in  series  between  a 
and  b,  the  electromotive  force  between  a  and  b  is  equal  to  Z/2  X  2^>n,  or 
abvolts. 


71.  The  induction  coil.f  —  An  iron  rod  wound  with  insulated 
wire  may  be  repeatedly  magnetized  and  demagnetized  by  con- 
necting a  battery  to  the  winding  and  repeatedly  making  and 
breaking  the  circuit.  The  increasing  and  decreasing  magnetic 

*  A  type  of  armature  winding  which  is  frequently  employed  provides  but  two  paths 
through  the  armature  winding  irrespective  of  the  number  of  field  magnet  poles. 

f  The  induction  coil  was  invented  by  Ruhmkorfi  in  1855  an(^  ^  1S  frequently  called 
the  Rukmkorff  coil. 


132        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

flux  thus  produced  through  the  rod  may  be  utilized  to  induce 
electromotive  force  in  an  auxiliary  coil  of  wire  wound  on  the  rod. 
Such  an  arrangement  13  called^an  induction  coil.  The  winding 
through  which  the  magnetizing  current  from  the  battery  flows  is 
called  the  primary  coil  and  the  auxiliary  winding  in  which  the 
desired  electromotive  force  is  induced  is  called  the  secondary  coil. 
The  iron  rod  is  always  made  of  a  bundle  of  fine  iron  wires  to  pre- 
vent the  flow  of  eddy  currents  as  explained  in  Art.  74. 

When  the  iron  rod  or  core  is  magnetized  a  pulse  of  electromo- 
tive force  is  induced  in  the  secondary  coil,  and  when  the  iron  core 
is  demagnetized,  a  reversed  pulse  of  electromotive  force  is  induced 
in  the  secondary  coil.  These  impulsive  electromotive  forces  may 
be  made  very  large  in  value,  hundreds  of  thousands  of  volts,  by 
making  the  secondary  coil  of  many  turns  of  wire  and  by  provid- 
ing for  the  quickest  possible  magnetization  and  demagnetization 
of  the  core. 

A  battery  or  any  ordinary  current  generator  does  not  magnetize 
a  core  very  quickly  when  connected  to  a  winding  of  wire ;  in 
fact,  a  very  considerable  fraction  of  a  second  is  usually  required 
for  the  core  to  become  magnetized.  Therefore,  during  the  mag- 
netization of  the  iron  core  of  an  induction  coil  the  electromotive 
force  induced  in  the  secondary  coil  is  a  comparatively  weak  pulse 
of  long  duration. 

On  the  other  hand,  proper  arrangements  permit  of  an  extremely 
quick  demagnetization  of  the  iron  core  of  an  induction  coil  when 
the  battery  is  disconnected  from  the  primary  winding,  and  this 
quick  demagnetization  induces  in  the  secondary  coil  an  intense 
pulse  of  electromotive  force  of  short  duration. 

The  quick  demagnetization  of  the  iron  core  of  an  induction 
coil  is  accomplished  as  follows  :  Figure  92*2  shows  the  connections 
of  a  battery  to  the  primary  coil.  The  battery  is  connected  and 
disconnected  by  making  and  breaking  contact  between  the  metal 
terminals  tt.  Two  large  metal  plates  separated  by  an  insulator 
(a  condenser)  are  connected  to  the  terminals  //  as  shown. 
When  the  points  tt  are  connected  together  the  core  is  slowly 


INDUCED  ELECTROMOTIVE  FORCE. 


133 


magnetized  by  the  current  from  the  battery.     When  the  points 
tt  are  separated,  the  current  persists  in  flowing  for  a  short  in- 


iron  8 


core 


battery 


Fig.  92a. 

terval  of  time,  this  persisting  current  flows  into  the  condenser 
plates,  and  the  electric  charge  which  thus  accumulates  on  the 
plates  surges  back  through 
the  circuit  as  a  reversed  cur- 
rent and  demagnetizes  the 
iron  core. 

Figure  92^  shows  a  com- 
plete induction  coil.  The 
condenser  is  mounted  inside 
of  the  box-like  base.  Fig  92b 

72.  The  alternating-current  transformer  consists  of  two  coils  of 
wire,  a  primary  coil  and  a  secondary  coil,  wound  upon  an  iron 
core.  This  iron  core  is  built  up  of  strips  of  sheet  iron,  and  it 
usually  forms  a  complete  magnetic  circuit  as  shown  in  Fig.  93. 
Figure  94  shows  a  sectional  view  of  Fig.  93  ;  the  primary  coil  is 
represented  by  PP  and  a  secondary  coil  by  55.  Either  coil 
of  a  transformer  may  be  the  primary  coil  according  to  the  way  in 
which  the  transformer  is  used  as  explained  later.  The  induction 
coil  and  the  alternating-current  transformer  are  identical,  except 


134        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


that  the  iron  core  of  the  induction  coil  is  not  a  complete  magnetic 
circuit,  but  has  magnet  poles  at  its  ends.     The  effect  of  these 


:••! 


iron   —=— 


— .   iron 


Fig.  93. 


Fig.  94. 


magnet  poles  is  to  facilitate  the  quick  demagnetization  of  the  core 
when  the  primary  circuit  of  the  induction  coil  is  broken. 

The  action  of  the  transformer. — Alternating  current  is  supplied 
to  either  coil  of  the  transformer.  This  alternating  current  pro- 
duces rapid  reversals  of  magnetization  of  the  iron  core.  These 
magnetic  reversals  induce  an  alternating  electromotive  force  in 
the  other  coil  which  delivers  alternating  current  to  any  circuit  to 
which  it  may  be  connected.  The  coil  of  a  transformer  which  re- 
ceives alternating  current  is  called  the  primary  coil,  and  the  coil 
which  delivers  alternating  current  is  called  the  secondary  coil. 

Step-up  and  step-down  transformation. — Usually,  one  coil  of  a 
transformer  has  many  more  turns  of  wire  than  the  other.  The 
coil  of  many  turns  may  act  as  the  primary  coil,  taking  a  small 
current  at  high  electromotive  force  from  an  alternator ;  and  in 
this  case  the  coil  of  few  turns  will  be  the  secondary  coil,  and  it 
will  deliver  a  large  current  at  low  electromotive  force  to  a  receiv- 
ing circuit.  This  action  is  called  step-down  transformation. 

The  coil  of  few  turns  on  the  other  hand  may  act  as  the  primary 
coil,  taking  a  large  current  at  low  electromotive  force  from  an 
alternator;  and  in  this  case  the  coil  of  many  turns  will  be  the 
secondaiy  coil  and  it  will  deliver  a  small  current  at  high  voltage 
to  a  receiving  circuit.  This  action  is  called  step-up  transformation. 

The  object  of  step-up  and  step-down  transformation  may  be 
explained  as  follows :  The  transmission  of  a  given  amount  of 


INDUCED   ELECTROMOTIVE   FORCE.  135 

power  electrically  may  be  accomplished  by  transmitting  the  large 
current  output  of  a  low  voltage  generator,  or  by  transmitting  the 
small  current  output  of  a  high  voltage  generator.  In  the  former 
case  very  large  and  expensive  transmission  wires  must  be  used  or 
the  loss  of  power  in  the  transmission  wires  will  be  excessive.  In 
the  latter  case,  comparatively  small  and  inexpensive  transmission 
wires  may  be  used  without  involving  an  excessive  loss  of  power. 
Therefore  high  electromotive  force  is  a  practical  necessity  in  the 
long  distance  transmission  of  power.  The  user  of  electric  power 
must  however  be  supplied  with  current  at  low  electromotive 
force,  partly  on  account  of  the  danger  involved  in  the  use  of  high 
electromotive  forces,  and  partly  on  account  of  the  fact  that  many 
types  of  electrical  apparatus  cannot  be  operated  satisfactorily  with 
high  electromotive  force ;  also  it  is  inconvenient  and  dangerous  to 
generate  very  high  electromotive  forces  in  a  complicated  machine 
like  an  alternator  which  must  be  cared  for  by  an  attendant. 
These  difficulties  are  met  by  employing  a  transformer  for  step-up 
transformation  at  the  generating  station  and  another  transformer 
for  step-down  transformation  at  the  receiving  station. 

High  efficiency  of  the  transformer.  —  The  transformer  is  not  only 
cheap  to  construct  and  cheap  to  operate,  but  it  is  extremely 
efficient.  The  efficiency  ranges  from  95  or  96  per  cent,  for 
small  sized  transformers  to  98  per  cent,  or  more  for  transformers 
of  large  size. 

73.  Current  and  electromotive  force  relations  of  the  transformer  — In  the 

following  discussion  Zf  represents  the  number  of  turns  of  wire  in  the  primary  coil, 
and  Z"  the  number  of  turns  of  wire  in  the  secondary  coil.  The  effect  of  the  elec- 
trical resistance  of  the  coils,  which  is  usually  quite  small  in  practice,  is  ignored,  and 
all  of  the  magnetic  flux  which  passes  through  one  coil  is  assumed  to  pass  through  the 
other  coil  also. 

(a)  Electromotive  force  relations. — Let  Ef  be  the  effective  value  of  the  alter- 
nating electromotive  force  which  acts  on  the  primary  coil  of  a  transformer,  and  let  E" 
be  the  effective  value  of  the  electromotive  force  induced  in  the  secondary  coil  of  the 
transformer.  Then 

E'        Zf 

~E»=^»  (46) 

This  relation  may  be  shown  to  be  true  as  follows  :  The  only  thing  which  opposes  the 
flow  of  current  through  the  primary  coil  is  the  reacting  electromotive  force  induced  in 


136        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  primary  coil  by  the  reversals  of  magnetization  of  the  core  (resistance  of  primary 
coil  being  neglected).  Therefore,  the  electromotive  force  which  acts  upon  the  primary 
coil  is  equal  and  opposite  to  the  electromotive  force  which  is  induced  in  the  primary 
coil  by  the  magnetic  reversals  of  the  core.  The  magnetic  reversals  of  the  core,  how- 
ever, induce  a  certain  electromotive  force  f  in  each  turn  of  wire  surrounding  the  core. 
Therefore  the  total  electromotive  force  induced  in  the  primary  coil  is  Z'e  and  the 
total  electromotive  force  induced  in  the  secondary  coil  is  Z"e  so  that  the  ratio  of  the 
two  electromotive  forces  is  equal  to  Z'/Z". 

(b)  Current  relations.  —  The  electromotive  force  which  Is  induced  in  the  primary 
coil  of  a  transformer  balances  the  electromotive  force  which  is  applied  to  the  primary 
coil  as  explained  above,  and  the  range  of  reversals  of  magnetization  of  the  core  must 
be  such  as  to  induce  this  reacting  electromotive  force  in  the  primary  coil.  There- 
fore the  combined  magnetizing  action  of  primary  and  secondary  coils  is  always  such 
as  to  magnetize  the  core  to  that  degree  which  will  make  the  reacting  electromotive 
force  in  the  primary  coil  equal  to  the  electromotive  force  of  the  alternator  which  is 
forcing  current  through  the  primary  coil. 

When  the  secondary  coil  is  on  open  circuit,  just  enough  current  flows  through  the 
primary  coil  to  produce  the  degree  of  magnetization  above  specified.  Let  this  value 
of  the  primary  current,  which  is  called  the  magnetizing  current  of  the  transformer, 
be  represented  by  /'.  When  a  current  1"  is  taken  from  the  secondary  coil  a  current 
If  in  addition  to  the  magnetizing  current  i  flows  through  the  primary  coil.  The 
current  i  still  suffices  to  magnetize  the  core,  and  the  magnetizing  action  of  Iff  is  ex- 
actly neutralized  by  the  equal  and  opposite  magnetizing  action  of  P  '.  The  magnetiz- 
ing action  of  Ff  is  measured  by  the  product  Zff  I"  and  the  magnetizing  of  I'  is 
measured  by  the  product  ZV,  so  that,  ignoring  algebraic  signs,  we  have 

Z'F  =  Z»l" 

or 

••-•  If        Z" 


74.  Eddy  currents.  Lamination.  —  When  an  iron  rod  is  mag- 
netized or  demagnetized,  the  changing  magnetic  flux  through  the 
central  portions  of  the  rod  induces  electromotive  forces  around 
the  outer  portions  of  the  rod,  and  these  electromotive  forces  pro- 
duce what  are  called  eddy  currents.  Eddy  currents  are  also 
produced  in  a  mass  of  metal  which  is  near  to  a  moving  magnet 
or  which  moves  in  the  neighborhood  of  a  stationary  magnet. 

Lamination.  —  Those  parts  of  electrical  machinery  which  are 
subject  to  rapid  and  frequent  changes  of  magnetization  are  always 
built  up  of  iron  wire  or  of  thin  sheets  of  iron  so  as  to  leave  the 
iron  continuous  in  the  direction  of  the  magnetization  but  discon- 
tinuous in  the  direction  in  which  the  eddy  currents  tend  to  flow. 


INDUCED   ELECTROMOTIVE   FORCE.  137 

Such  a  mass  of  iron  is  said  to  be  laminated.  The  iron  parts  of 
dynamo  armatures  and  of  transformers  are  always  laminated. 

Examples  of  eddy  currents.  —  A  suspended  magnet  which  is 
set  oscillating  about  its  axis  of  suspension  is  quickly  brought  to 
rest  if  it  is  surrounded  by  a  massive  ring  of  copper,  because  the 
eddy  currents  induced  in  the  copper  by  the  moving  magnet  act 
upon  the  magnet  with  a  force  which  is  at  each  instant  opposed  to 
the  motion  (Lenz's  Law). 

A  sheet  of  copper  which  is  suddenly  thrust  between  the 
poles  of  a  strong  electromagnet  behaves  as  if  it  were  moving 
in  a  viscid  liquid.  Eddy  currents  are  induced  in  the  copper 
and,  because  of  these  eddy  currents,  the  magnet  exerts  a  force 
upon  the  copper  which  is  always  opposed  to  the  motion  (Lenz's 
Law). 

An  interesting  effect  of  eddy  currents  is  their  action  in  prevent- 
ing the  sudden  magnetization  or  demagnetization  of  a  solid  iron 
rod.  Thus,  a  bundle  of  iron  wires  surrounded  by  a  winding  of 
wire  is  magnetized  say  in  one  second  when  the  winding  is  con- 
nected to  a  given  battery,  and  demagnetized  in  a  much  shorter  time 
when  the  battery  is  disconnected.  A  solid  iron  rod  of  the  same 
size  would  require  perhaps  nine  or  ten  seconds  to  be  magnetized 
by  the  same  coil  and  battery,  and  the  solid  rod  would  lose  its 
magnetism  very  slowly  when  the  battery  is  disconnected.  The 
eddy  currents  in  the  solid  rod  oppose  the  magnetization  while  the 
rod  is  being  magnetized,  and  they  tend  to  keep  up  the  magnetization 
while  the  rod  is  being  demagnetized  (Lenz's  Law).  Another  in- 
teresting effect  of  eddy  currents  is  that  which  is  exemplified  in  the 
ordinary  "medical"  induction  coil,  in  which  the  "  power"  of  the 
coil  is  adjusted  by  moving  a  brass  or  copper  tube  which  surrounds 
the  iron  core  of  the  coil.  When  the  tube  surrounds  the  entire 
core  a  sudden  break  in  the  primary  circuit  results  in  a  slow  de- 
magnetization of  the  core  because  of  the  eddy  currents  in  the  tube 
which  tends  to  keep  up  the  magnetization,  but  when  the  tube  is 
withdrawn  the  core  is  demagnetized  very  quickly  when  the 
primary  circuit  is  broken. 


138        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

PROBLEMS. 

92.  Let  it  be  assumed  that  the  force  required  to  propel  a  canal 
boat  is  proportional  to  the  velocity  of  the  boat.     A  force  of  50 
pounds  is  required  to  maintain  "a  velocity  of  5  feet  per  second. 
(a)  Find  the  value  of  the  coefficient  by  which  the  velocity  of  the 
boat  must  be  multiplied  to  give  the  frictional  drag  and  specify  the 
unit  in  terms  of  which  this  coefficient  is  expressed,     (b)  Find  the 
velocity  at  which  the  boat  would  be  propelled  by  a  force  of  36 
pounds,     (c)  Find  the  rate  at  which  work  is  done  by  a  force  of 
36   pounds   in    propelling  the   boat.     Ans.  (a)   10  pounds  per 
(foot  per  second),     (b)  3.6  feet/second,     (c)   129.6  foot-pounds/ 
second. 

93.  When  a  force  of  50  pounds  is  applied  to  the  above  canal 
boat  the  boat  starts  from  rest  and  after  some  time  reaches  its  full 
speed  of  5  feet- per  second.    At  a  given  instant  the  velocity  of  the 
boat  is  3  feet  per  second.     At  this  instant :  (a)  Find  the  rate  at 
which  work  is  done  on  the  boat  by  the  propelling  force,    (ft)  Find 
the  dragging  force  which  is  acting  on  the  boat,     (c)  Find  the  rate 
at  which  work  is  dissipated  in  overcoming  the  friction  of  the  water. 
(d)  Explain  what  is  becoming  of  the  difference  between  (a)  and 
(c).    Ans.  (a)   150  foot  pounds/second.     (b)  30  pounds,    (c)  90 
foot-pounds/second. 

94.  An  electromotive  force  of  50  volts  acts  on  a  circuit  of  which 
the  resistance  is  10  ohms.     At  a  certain  instant  during  the  time 
that  the  current  is  growing  from  zero  to  its  full  value  the  current  has 
an  actual  value  of  3  amperes.     At  this  instant :  (a)  Find  the  rate 
at  which  the  generator  delivers  work  to  the  circuit,     (b)  Find  the 
dragging  force  in  volts  which  is  opposing  the  flow  of  the  current 
through  the  circuit,     (c)  Find  the  rate  at  which  work  is  dissipated 
in  overcoming  the  resistance  of  the  circuit,     (d)  Explain  what  is 
becoming  of  the  difference  between  (a)  and  (c).     Ans.   (a)   150 
watts,     (b)  30  volts,     (c)  go  watts. 

95.  A  vertical  wire  3  meters  long  is  moved  sidewise,  towards 
magnetic  east  or  west,  at  a  velocity  of  25  meters  per  second. 


INDUCED   ELECTROMOTIVE   FORCE.  139 

Find  the  electromotive  force  induced  in  the  wire  in  volts,  the 
horizontal  component  of  the  earth's  field  being  o.  1 8  gauss. 
Ans.  0.00135  volt. 

96.  The  pole-face  of  a  dynamo  is  30  centimeters  long  in  the 
direction  parallel  to  the  axis  of  the  armature,  and  the  field  inten- 
sity in  the  gap  space  between  the  pole-face  and  the  armature  core 
is  6,000  gausses.     The  wires  on  the  armature  are  1 2  centimeters 
from  the  axis  of  the  armature,  and  the  speed  of  the  armature  is 
1, 800  revolutions  per  minute.     Find  the  electromotive  force  in 
volts  induced  in  each  armature  wire  (30  centimeters  in  length)  as 
it  passes  across  the  pole-face.     Ans.  4.07  volts. 

97.  A  single  wire    Wy    Fig..  83,   is  rotated  along  the  dotted 
line  in  Fig.  83  at  a  speed  of  25   revolutions  per  second.     The 
magnetic  flux  which  emanates  from  each  north  pole  of  the  field 
magnet  and  which  enters  each  south  pole  is  2,500,000   lines. 

(a)  Find  the  average  value  of  the  electromotive  force  induced  in 
the  wire  during  the  time  that  it  sweeps  from  a  point  midway  be- 
tween two  field  poles  to  the  next  point  midway  between  two 
poles,     (fr)  Find  the  number  of  cycles  per  second  through  which 
this  induced   electromotive   force  passes.      Ans.    (a)   2.5  volts. 

(b)  50  cycles  per  second. 

98.  The  alternator  specified  in  problem  97  has  a  winding  as 
shown  in  Fig.  84.     Find  the  average  value  of  the  electromotive 
force  induced  in  the  winding  during  the  time  that  the  armature 
is  making  one  fourth  of  a  revolution  (that  is,  during  the  time  that 
the  slots  containing  the  wires  travel  from  a  point  midway  between 
the  pole  pieces  to  another  set  of  points  midway  between  the  pole 
pieces.     Ans.    10  volts. 

99.  The  core  of  an  induction  coil  carries   100,000  lines    of 
magnetic  flux,  when  current  is  flowing  through  the  primary  coil. 
When  the  primary  circuit  is  broken  the  flux  in  the  core  drops  to 
T  0,000  lines  in  0.003  second.      How  many  turns  of  wire  are  re- 
quired in  the  secondary  coil  in  order  that  an  average  electromo- 
tive force  of  25,000  volts  may  be  induced  in  this  coil  during  the 
0.003  second?     Ans.   83,333  turns. 


140        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

100.  The  ring  armature  of  a  direct-current  bipolar  dynamo  has  260  turns  of  wire 
upon  it,  the  armature  is  driven  at  a  speed  of  1,200  revolutions  per  minute,  and  the 
magnetic  flux  from  a  pole-face  into  the  armature  core  is  2,200,000  lines.     Calculate 
the  electromotive  force  of  the  dynamo  in  volts.     Ans.    114.4  volts. 

101.  The  armature  described  in  the  above  problem  has  upon 
it  500  feet  of  pure  copper  wire  325  mils  in  diameter.     What  is 
the   resistance   of  the  armature   from   brush  to   brush  ?      Ans. 
0.0125  ohm. 

Remark.  —  In  a  bipolar  dynamo  the  wire  on  the  armature  constitutes  two  paths 
between  the  brushes. 

102.  A  transformer  takes  alternating  current  from  supply  mains  at  I,IOO  volts  and 
delivers  current  to  service  mains  at   no  volts.     The  primary  coil  of  the  transformer 
has  560  turns  of  wire.     How  many  turns  of  wire  are  there  in  the  secondary  coil  ? 
The  transformer  delivers  250  amperes  to  the  service  mains.     How  much  current  does 
it  take  from  the  supply  mains?     A  usual  allowance  in  transformer  coils  is  1,000  cir- 
cular mils  sectional  area  of  wire  for  each  ampere.     Find  size  of  wire  used  in  primary 
coil  and  in  secondary  coil  of  the  transformer.     Ans.  56  turns,  25  amperes,  25,000 
cir.  mils,  250,000  cir.  mils. 


CHAPTER   VI. 
ELECTRIC   MOMENTUM.     INDUCTANCE. 

75.  The  momentum  of  the  electric  current.     Spark  at  break.  — 

The  analogy  between  electric  current  strength  and  velocity,  as 
outlined  in  Art.  62,  would  lead  one  to  expect  an  electric  current 
to  possess  momentum  and  kinetic  energy  very  much  as  a  moving 
body  possesses  momentum  and  kinetic  energy.  In  fact,  this  is 
found  to  be  the  case.  When  an  electric  circuit  is  broken,  the 
current  continues  to  flow  across  the  break  for  a  short  time,  pro- 
ducing an  electric  arc  or  spark,  and  the  intensity  of  this  spark  is 
a  rough  indication  of  the  amount  of  kinetic  energy  possessed  by 
the  current. 

The  amount  of  kinetic  energy  associated  with  a  given  current  in 
a  circuit  made  of  a  given  length  and  size  of  wire,  depends  upon  the 
shape  of  the  circuit  and  upon  the  presence  of  iron  near  the  circuit. 
Thus,  a  current  in  circuit  a,  Fig.  95,  possesses  but  little  kinetic 


Iron 


energy ;  the  same  current  in  circuit  b  possesses  more  kinetic 
energy ;  and  the  same  current  in  circuit  c  possesses  very  much 
more  kinetic  energy.  When  the  circuit  of  an  ordinary  incan- 
descent lamp  is  broken  a  very  slight  spark  only  is  produced ;  a 

141 


142        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

coil  of  wire  having  the  same  resistance  as  the  lamp  is  connected 
to  the  supply  mains  so  as  to  take  the  same  amount  of  current  as 
the  lamp,  and  a  much,  more  intense  spark  is  produced  when  this 
circuit  is  broken  ;  an  iron  core  consisting  of  a  bundle  of  iron  wires 
is  then  placed  in  the  coil  and  a  spark  several  inches  in  length  may 
be  produced  by  suddenly  breaking  the  circuit^. 

The  kinetic  energy  of  the  electric  current  resides  in  the  mag- 
netic field  which  is  produced  by  the  current.  Thus,  a  current  in 
the  circuit  a,  Fig.  95,  produces  a  very  weak  magnetic  field  except 
in  the  small  region  between  the  wires,  and  the  kinetic  energy  of 
the  current  is  small.  The  same  current  in  the  circuit  b  produces 
an  intense  magnetic  field  inside  of  the  coil  and  the  kinetic  energy 
of  the  current  is  correspondingly  great.  The  kinetic  energy  of 
the  current  in  the  coil  of  wire  shown  in  Fig.  g$c  is  much  greater 
than  the  kinetic  energy  of  the  same  current  in  the  circuit  shown 
in  Fig.  95^,  but  the  presence  of  the  iron  core  in  Fig.  95$  com- 
plicates matters  greatly,  and  nearly  the  whole  of  this  chapter 
relates  to  the  kinetic  energy  of  currents  in  the  absence  of  iron. 

Practical  applications  of  the  spark  at  break.  —  In  the  device 
which  is  ordinarily  used  for  lighting  gas  jets  by  electricity,  an 
electric  circuit  is  made  and  broken  in  the  stream  of  gas  which  is 
to  be  lighted,  and  the  gas  is  ignited  by  the  spark  at  break.  In 
order  to  produce  an  intense  spark,  the  circuit  includes  a  coil  of 
wire  wound  on  an  iron  wire  core,  a  so-called  "  spark  coil."  This 
same  device  is  used  for  igniting  the  mixture  of  gas  and  air  in  a 
gas  engine. 

76.  Definition  of  inductance.  —  The  kinetic  energy  which  is 
associated  with  a  current  in  a  given  circuit  is  proportional  to  the 
square  of  the  current ;  that  is,  we  may  write 

W~  \LP  (48) 

in  which  W  is  the  kinetic  energy  of  a  current  /  in  a  given  cir- 
cuit, and  (JZ)  is  the  proportionality  factor.  The  quantity  L  is 
called  the  inductance*  of  the  circuit. 

*  Sometimes  called  the  coefficieni  of  self-induction  of  the  circuit. 


ELECTRIC    MOMENTUM.     INDUCTANCE.  143 

Discussion  of  equation  (48).  —  It  was  pointed  out  in  Art.  53 
that  to  double  the  strength  of  the  current  in  a  circuit  is  to  double 
everywhere  the  intensity  of  the  magnetic  field  which  is  due  to  the 
current,  and  it  was  shown  in  Art.  44  that  the  kinetic  energy  per 
unit  volume  of  a  magnetic  field  is  proportional  to  the  square  of 
the  field  intensity.  Therefore  to  double  the  strength  of  the  cur- 
rent in  a  circuit  is  to  double  everywhere  the  intensity  of  the  mag- 
netic field  due  to  the  current,  and  to  quadruple  everywhere  the 
energy  of  the  magnetic  field,  so  that  to  double  an  electric  current 
is  to  quadruple  the  total  energy  of  its  magnetic  field. 

Units  of  inductance.  —  If  W  in  equation  (48)  is  expressed  in 
joules,  and  /  in  amperes,  then  L  is  expressed  in  terms  of  a  unit 
of  inductance  which  is  called  the  henry ',  that  is  to  say,  a  circuit 
has  an  inductance  of  one  henry  when  a  current  of  one  ampere  in 
this  circuit  represents  one  half  of  a  joule  of  kinetic  energy. 

If  W  in  equation  (48)  is  expressed  in  ergs  and  /  in  abam- 
peres,  then  L  is  expressed  in  c.g.s.  units  of  inductance.  The 
c.g.s.  unit  of  inductance  is  sometimes  called  the  abhenry*  A  cir- 
cuit has  one  abhenry  of  inductance  when  a  current  of  one  abam- 
pere  in  that  circuit  represents  one  half  of  an  erg  of  kinetic  energy. 
There  are  io9  abhenrys  in  one  henry. 

Inductance  of  a  coil.  —  Strictly,  one  cannot  speak  of  the  in- 
ductance of  anything  but  an  entire  circuit,  inasmuch  as  every 
portion  of  a  circuit  contributes  its  share  to  the  magnetic  field  at 
each  and  every  point  in  the  surrounding  region ;  it  is,  however, 
allowable  to  speak  of  the  inductance  of  a  coil  when  the  terminals 
of  the  coil  are  not  too  far  apart,  and  when  the  remainder  of  the 
electrical  circuit  does  not  produce  any  perceptible  magnetic  field 
in  the  region  occupied  by  the  coil. 

Non-inductive  circuits.  —  A  circuit  is  said  to  be  non-inductive 
when  the  inductance  of  the  circuit  is  negligibly  small,  that  is, 
when  the  electromotive  force  L  x  dijdt^  is  negligibly  small  as 
compared  with  the  electromotive  force  RT  which  overcomes  the 

*The  c.g.s.  unit  of  inductance  is  often  called  the  centimeter. 
f  See  next  article. 


144        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

resistance  of  the  circuit.  Thus,  a  given  circuit  might  be  consid- 
ered to  be  non-inductive  under  conditions  involving  slow  changes 
of  current,  whereas  the  same  circuit  would  not  be  considered  to 
be  non-inductive  under  conditions  involving  rapid  changes  of  cur- 
rent. When  a  circuit  consists  simply  of  out-going  and  returning 
wires,  side  by  side,  its  inductance  is  so  small  that  it  may  be  in 
most  cases  ignored.  The  wires  used  in  resistance  boxes  are 
usually  arranged  non-inductively.  This  may  be  done  by  doubling 
the  wire  back  on  itself,  and  winding  this  doubled  wire  on  a  spool ; 
or  the  wire  may  be  wound  in  one  layer  on  a  thin  paper  cylinder, 
and  this  cylindrical  coil  may  then  be  flattened  so  as  to  reduce  the 
region  (inside)  in  which  the  magnetic  field  is  intense. 

Measurement  of  inductance.  —  The  most  accurate  method  for 
determining  the  inductance  of  a  coil  is  by  calculation  from  meas- 
ured dimensions.  This  calculation  can  be  carried  out  only  when 
the  coil  is  very  simple  in  shape,  and  even  then  the  calculation  is 
in  most  cases  quite  complicated.*  The  simplest  case  is  given  in 
Art,  8 1.  The  inductance  of  an  irregularly-shaped  coil  may  be 
determined  by  various  electrical  methods,  f 

Moment  of  inertia  of  a  wheel.  Analogue  of  inductance.  —  The 
kinetic  energy  of  a  rotating  wheel  resides  in  the  various  moving 
particles  of  a  wheel,  in  the  same  way  that  the  kinetic  energy  of  a 
current  resides  in  the  various  parts  of  the  magnetic  field  which-  is 
due  to  the  current.  If  the  angular  velocity  to  of  the  wheel  is 
doubled  the  linear  velocity  of  every  particle  of  the  wheel  is 
doubled,  in  the  same  way  that  the  intensity  of  the  magnetic  field 
at  every  point  in  the  neighborhod  of  a  coil  is  doubled  when  the 
current  in  a  coil  is  doubled.  Therefore  the  kinetic  energy  of 
every  particle  of  a  wheel  is  quadrupled  when  its  angular  velocity 
is  doubled,  in  the  same  way  that  the  kinetic  energy  of  every  por- 
tion of  the  magnetic  field  around  a  coil  is  quadrupled  when  the 

*  See  a  series  of  articles  in  the  Bulletin  of  the  United  States  Bureau  of  Standards, 
by  E.  B.  Rosa,  Vol.  I,  pages  125  and  291  ;  Vol.  2,  pages  87,  161  and  359 ;  Vol.  3, 
page  i. 

|  See  Practical  Physics,  by  Franklin,  Crawford  and  MacNutt,  Vol.  2,  page  129  ; 
see  also  Absolute  Measurements,  by  Andrew  Gray,  Vol.  2,  Part  2,  pages  438-509. 


ELECTRIC   MOMENTUM.     INDUCTANCE.  145 

current  in  the  coil  is  doubled  ;  and  consequently  the  total  kinetic 
energy  of  a  rotating  wheel  is  proportional  to  the  square  of  its 
angular  velocity,  in  the  same  way  that  the  total  kinetic  energy 
of  a  current  in  a  given  coil  is  proportional  to  the  square  of  the 
current.  That  is,  we  may  write 


in  which  W  is  the  kinetic  energy  of  a  rotating  wheel,  «  is  the 
angular  velocity  of  the  wheel,  and  (J-AT)  is  a  proportionality 
factor.  The  quantity  K  is  called  the  moment  of  inertia  of  the 
wheel. 

77.  Electromotive  force  required  to  cause  a  current  to  increase 
or  decrease.  —  To  maintain  a  constant  current  in  a  circuit  an  elec- 
tromotive force  equal  to  Ri  must  act  upon  the  circuit  to  over- 
come the  resistance  of  the  circuit.  If  the  electromotive  force 
which  acts  upon  the  circuit  is  greater  than  Ri,  the  current  in- 
creases in  value,  and  if  the  electromotive  force  which  acts  upon 
the  circuit  is  less  than  Ri,  the  current  decreases  in  value.  Let 
the  electromotive  force  which  acts  upon  a  circuit  exceed  Ri  by 
the  amount  e  ;  then  we  have 


in  which  L  is  the  inductance  of  the  circuit  and  difdt  is  the 
rate  at  which  the  current  increases.  When  e  is  negative  (elec- 
tromotive force  less  than  Ri)  then  difdt  is  negative,  that  is, 
the  current  decreases. 

Mechanical  analogue  of  equation  (49).  —  To  keep  a  body  in 
uniform  motion  a  force  sufficient  to  overcome  the  drag  of  friction 
must  act  upon  the  body.  If  the  force  which  acts  upon  the  body 
is  greater  than  the  drag  of  friction,  the  body  gains  velocity,  and 
if  the  force  which  acts  upon  the  body  is  less  than  the  drag  of  fric- 
tion, the  body  loses  velocity.  Let  the  force  which  acts  upon  the 
body  exceed  the  drag  of  friction  by  the  amount  e,  then  we  have 

di 


ii 


146        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

in  which  L  is  the  mass  of  the  body  and  dijdt  is  the  rate  at 
which  its  velocity  changes.  Equation  (49)  is  therefore  analogous 
to  the  fundamental  equation  in  mechanics  which  expresses  the 
relationship  between  unbalanced  force,  mass  and  acceleration. 

Starting  from  the  fact  that  force  equals  mass  times  accelera- 
tion, it  can  be  shown  that  the  kinetic  energy  of  a  moving  body  is 
equal  to  one-half  its  mass  times  its  velocity  squared,  suitable 
units  being  employed.  The  same  argument  reversed  would  show 
that  force  must  be  equal  to  mass  times  acceleration  if  kinetic 
energy  is  equal  to  one-half  mass  times  velocity  squared  ;  and  an 
exactly  similar  argument  would  establish  equation  (4.9)  on  the  basis 
of  equation  (4$). 

Self  -induced  electromotive  force.  —  When  one  pushes  on  a  body 
causing  its  velocity  to  increase  the  body  reacts  and  pushes  back 
on  the  hand.  This  reacting  force  is  equal  and  opposite  to  the 
acting  force  which  is  causing  the  increase  of  velocity.  When  the 
velocity  of  the  body  is  increasing,  its  reaction  is  a  force  opposed 
to  its  motion,  and,  when  the  velocity  of  the  body  is  decreasing, 
its  reaction  is  a  force  in  the  direction  of  its  motion. 

Similarly  when  an  electromotive  force  acts  upon  a  circuit  and 
causes  the  current  to  increase  or  decrease,  the  changing  current 
reacts,  and  the  reacting  electromotive  force  is  equal  and  opposite 
to  the  acting  electromotive  force  which  is  causing  the  current  to 
change.  Therefore  from  equation  (49)  we  have 


in  which  e  is  the  reaction  of  the  changing  current  in  a  circuit  of 
which  L  is  the  inductance,  and  dijdt  is  the  rate  at  which  the 
current  is  changing.  This  reaction  of  a  changing  current  is 
called  self-induced  electromotive  force. 

78.  Growth  of  current  in  an  inductive  circuit.  —  A  steady  force 
E  begins  to  act  upon  a  boat  at  a  given  instant,  starting  it  from 
rest.  At  the  given  instant  the  velocity  of  the  boat  is  zero,  the 
frictional  drag  of  the  water  is  zero,  and  all  of  the  force  is  used  to 


ELECTRIC    MOMENTUM.     INDUCTANCE.  147 

cause  the  velocity  of  the  boat  to  increase.  As  the  boat  gains 
more  and  more  velocity,  however,  a  larger  and  larger  portion 
of  the  force  E  is  used  to  overcome  the  frictional  drag  of  the 
water,  and  a  smaller  and  smaller  portion  of  E  is  used  to 
cause  the  velocity  of  the  boat  to  increase.  Finally,  after  the 
force  has  been  acting  for  some  time,  the  boat  reaches  full  speed, 
and  then  all  of  the  force  E  is  used  to  overcome  the  frictional 
drag. 

An  electromotive  force  E  due  to  a  battery  or  dynamo  begins 
to  act  on  a  circuit  at  a  given  instant.  At  this  instant  the  current 
is  zero,  and  the  whole  of  E  acts  to  cause  the  current  to  increase 
in  accordance  with  equation  (49).  As  the  increasing  current 
reaches  larger  and  larger  values,  however,  a  larger  and  larger 
portion  of  E  is  used  to  overcome  the  resistance  of  the  circuit, 
and  a  smaller  and  smaller  portion  of  E  is  used  to  cause  the 
current  to  increase.  After  the  electromotive  force  has  been  act- 
ing for  some  time  the  current  reaches  its  full  steady  value,  and 
then  the  whole  of  E  is  used  to  overcome  resistance.  The  por- 
tion of  E  which  is  used  at  any  given  instant  to  overcome  resist- 
ance is  equal  to  Ri  and  the  portion  which  is  used  to  cause  the 
current  to  increase  is  equal  to  L'difdf.  Therefore  we  have 


in  which  i  is  the  value  of  the  growing  current  at  a  given  instant, 
and   dijdt  is  its  rate  of  increase  at  that  instant. 

Examples.  —  (a)  A  force  of  50  pounds  propels  a  canal  boat  at 
a  speed  of  5  feet  per  second.  Let  us  assume  that  the  drag  of  the 
water  is  proportional  to  the  velocity  of  the  boat,  and  let  us  con- 
sider what  takes  place  during  the  time  that  the  boat  is  being 
started  from  rest  by  a  steady  force  of  50  pounds,  the  mass  of  the 
boat  being  100  tons.  At  the  very  start,  when  the  velocity  of 
the  boat  is  zero,  the  drag  of  the  water  is  zero,  and  the  propelling 
force  of  50  pounds  is  used  solely  to  produce  acceleration;  there- 
fore, from  the  formula  F  =-£-%-  ma,  we  find  the  acceleration  a 


148        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

to  be  0.008  foot  per  second  per  second.  After  the  boat  has 
gained  a  certain  amount  of  velocity,  say,  3  feet  per  second,  the 
drag  of  the  water  is.  J  of  50  pounds,  so  that  30  pounds  of  the 
propelling  force  is  used  to  overcome  the  drag  of  the  water  and 
the  remainder,  20  pounds,  is  used  to  produce  acceleration. 
Therefore  the  acceleration  is  0.0032  foot  per  second  per  second. 
At  the  very  start  when  the  velocity  of  the  boat  is  zero  no  work 
is  being  done  upon  the  boat.  When  the  velocity  of  the  boat 
becomes  3  feet  per  second,  the  propelling  force  does  work  at  the 
rate  of  150  foot-pounds  per  second;  a  portion  of  this  power 
is  expended  in  overcoming  the  friction  of  the  water,  and  the 
remainder  goes  to  increase  the  kinetic  energy  of  the  moving  boat. 
The  portion  of  power  which  is  used  to  overcome  the  friction  of 
the  water  is  found  by  multiplying  the  velocity  of  the  boat  by  the 
portion  of  the  force  which  is  used  to  overcome  the  frictional  drag. 
This  gives  90  foot-pounds  per  second.  The  portion  of  the  power 
which  goes  to  increase  the  kinetic  energy  of  the  boat  is  found  by 
multiplying  the  portion  of  the  propelling  force  which  produces 
acceleration  by  the  velocity  of  the  boat.  This  gives  60  foot- 
pounds per  second. 

(fr)  An  electric  generator  has  an  electromotive  force  of  50  volts 
and  it  acts  upon  a  circuit  of  which  the  resistance  is  10  ohms,  so 
that  the  steady  current  that  may  be  produced  by  the  electro- 
motive force  is  5  amperes  according  to  Ohm's  law.  The  induc- 
tance of  the  circuit  is,  say,  2  henrys.  At  the  instant  when  the 
electromotive  force  begins  to  act  on  the  circuit  the  current  is 
zero,  and  all  of  the  electromotive  force  is  used  to  cause  the  cur- 
rent to  increase  so  that  the  rate  of  increase  of  the  current  is  25 
amperes  per  second,  according  to  equation  (49).  After  the  cur- 
rent has  reached  a  value  of,  say,  3  amperes,  a  portion  of  the 
electromotive  force  of  the  generator  is  used  to  overcome  the 
resistance  of  the  circuit  and  a  portion  is  used  to  cause  the  current 
to  increase.  The  electromotive  force  which  is  used  to  overcome 
the  resistance  is  found  by  multiplying  the  resistance  of  the  circuit 
by  the  current  which  gives  30  volts,  and  the  remaining  20  volts 


ELECTRIC   MOMENTUM.     INDUCTANCE. 


149 


cause  the  current  to  increase  at  a  rate  of  10  amperes  per  second 
in  accordance  with  equation  (49).* 

The  ordinates  of  the  curve  in  Fig.  96  represent  the  successive 


Growing  current 
R=3  ohms 
L=0.04  hemy 
volts 


haadredtbs  of  a  second 


23456  78 

Fig.  96. 

values  of  growing  current  in  a  circuit  of  which  the  resistance  is  3 
ohms  and  the  inductance  is  0.04  henry,  the  electromotive  force 
of  the  generator  being  1 1  o  volts.  The  equation  to  this  curve  of 
growing  current  is 

-R- 

(52) 


E 
R 


E        * 

.  _  .  .  p   -^ 

R 


in  which  e  is  the  Naperian  base,   i  is  the  value  of  the  growing 
current  t  seconds  after  the  electro- 
motive force  E  is  connected  to  the 

circuit,  L  is  the  inductance  of  the         

circuit  and  R  is  its  resistance.  I  m 


79.  Decay  of  current  in  an  indue-  -E 
tive  circuit.  —  A  current  /  is  es- 
tablished in  an  inductive  circuit,  a 
piece  of  metal  mm  is  then  laid 
across  the  terminals  of  the  circuit 
and  the  battery  disconnected  as  in- 
dicated by  the  dotted  line  in  Fig.  97. 


___ 

(..J 


Fig.  97. 


Under  these  conditions 

*  Ohm's  law  does  not  apply  to  a  circuit  unless  El  equals  RP  as  stated  in  Art.  19. 


150        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  current  decreases  at  such  a  rate  that  the  self-induced  elec- 
tromotive force  (reaction  of  the  changing  current)  —  L  •  dijdt  is 
equal  to  Ri.  This  condition  is  expressed  by  the  equation 

0=Ri+L^t  (53) 

t. 

and  this  equation  may  be  most  easily  interpreted  as  follows  :  The 
electromotive  force  which  is  acting  on  the  circuit  is  equal  to  zero 
and  this  electromotive  force  is  divided  into  the  two  parts,  Ri 
which  is  used  to  overcome  the  resistance  and  L  •  dijdt  *  which  is 
used  to  cause  the  current  to  change. 

Examples.  —  (a)  The  canal  boat  mentioned  in  example  (a)  of 
the  preceding  article  is  brought  up  to  a  speed  of  4  feet  per  second 
and  then  the  propelling  force  ceases  to  act.  The  drag  of  the 
water  is  of  course  equal  to  4/5  of  50  pounds  when  the  velocity 
is  4  feet  per  second,  and  this  dragging  force  of  40  pounds  pro- 
duces a  negative  acceleration  or  retardation  of  0.0064  foot  Per 
second  per  second.  The  rate  at  which  the  kinetic  energy  of  the 
boat  is  being  dissipated  in  overcoming  the  frictional  drag  of  the 
water  may  be  found  by  multiplying  the  frictional  drag  of  40 
pounds  by  the  velocity  of  4  feet  per  second  which  gives  160 
foot-pounds  per  second. 

(£)  A  current  of  4  amperes  is  established  in  the  circuit  which 
is  specified  in  example  (£)  of  the  preceding  article.  At  a  given 
instant  the  circuit  is  closed  on  itself  and  the  current  is  left  to  die 
away.  At  this  instant  the  value  of  Ri  is  40  volts,  that  is,  the 
electromotive  force  required  to  overcome  the  resistance  of  the 
circuit  when  the  current  is  4  amperes  is  40  volts  and  this  elec- 
tromotive force  comes  from  the  reaction  of  the  decreasing  current, 
so  that  the  current  must  be  decreasing  at  a  rate  of  20  amperes  per 
second  according  to  equation  (49).  The  rate  at  which  the  kinetic 
energy  of  the  current  is  being  dissipated  in  overcoming  the  resist- 
ance of  the  circuit  may  be  found  by  multiplying  the  value  of  Ri 

*  This  part  must  of  course  be  negative,  and  therefore  dijdt  is  negative,  that  is,  the 
current  i  is  decreasing. 


ELECTRIC    MOMENTUM.     INDUCTANCE.  !$! 

in  volts  by  the  value  i  in  amperes  (equals  Ri2)  which  gives  160 
watts. 

The  ordinates  of  the  curve  in  Fig.  98  represent  the  successive 
values  of  a  decaying  current  in  a  circuit  of  which  the  resistance 


Decaying  current 
R*=3  ohms 
If=O.04  henry 
.1-36.7  amperes 


hnndredths  of  a  second 


4        A 

Fig.  98 


is  3  ohms  and  the  inductance  is  0.04  henry,  the  initial  value  of 
the  current  being  36.7  amperes. 

The  equation  of  the  curve  of  decaying  current  in  Fig.  98  is 


*-/•«  z  (54)* 

in  which  e  is  the  Naperian  base,  /  is  the  value  of  the  decaying 
current  at  the  instant  from  which  time  is  reckoned,  and  i  is  the 
value  of  the  decaying  current  t  seconds  later. 

80,  The  choke  coil.  —  A  coil  having  considerable  inductance  is 
frequently  used  for  the  choking  of  rapid  fluctuations  of  current. 
Such  a  coil  is  called  a  choke  coil.  When  a  choke  coil  is  connected 
to  the  terminals  of  an  alternator  the  rapidly  alternating  electro- 
motive force  of  the  alternator  produces  but  little  current  through 
the  coil.  This  is  analogous  to  the  fact  that  a  rapidly  alternating 

*  Equations  (52)  and  (54)  are  obtained  by  integrating  the  differential  equations 
(51)  and  (53)  with  due  reference  to  the  known  value  of  the  current  at  the  instant 


152         ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


force  (a  force  which  is  repeatedly  reversed  in  direction)  produces 
but  little  to  and  fro  motion  of  a  heavy  body  even  though  the 
frictional  opposition  to  motion  be  negligibly  small. 

One  of  the  most  important  uses  of  the  choke  coil  is  in  connec- 
tion with  the  lightning  arrester.  Figure  99  represents  a  dynamo 
G  supplying  current  to  a  trolley  wire.  When  this  wire  is  struck 
by  lightning  a  sudden  rush  of  current  takes  place  through  G  to 
earth,  and  this  rush  of  current  may  prove  disastrous  to  the 
dynamo  by  breaking  through  the  insulation  instead  of  following 
the  windings  of  wire  in  the  machine.  By  placing  a  choke  coil 
C  in  the  position  shown  in  the  figure,  the  lightning  discharge  is 

trolley  wire 


ground 


Fig.  99. 


made  to  break  through  a  short  air  gap  g  and  flow  to  earth 
harmlessly.  When  the  air  gap  g  has  been  broken  down  in  this 
way,  that  is,  when  a  spark  or  arc  has  been  established  across  the 
gap,  it  is  a  good  conductor,  and  the  dynamo  G  is  short-circuited. 
Therefore  a  lightning  arrester  must  be  provided  with  an  arrange- 
ment for  stopping  the  flow  of  the  dynamo  current  across  the  gap 
g  after  the  rush  of  current  from  the  lightning  stroke  has  ceased. 
This  is  sometimes  done,  as  in  the  Thomson  arrester,  by  means 
of  a  strong  magnet  which  produces  an  intense  magnetic  field  in 
the  region  of  the  gap  and  pushes  the  arc  sidewise,  and  blows  it 
out.  This  device  is  called  the  magnetic  blow-out.  The  entire 
arrangement  of  a  choke  coil  C,  air  gap  g,  and  magnetic  blow- 


ELECTRIC    MOMENTUM.     INDUCTANCE. 


153 


out  (which  is  not  shown  in  the  figure)  constitute  what  is  called  a 
lightning  arrester.* 

Example.  —  A  coil  of  heavy  copper  wire  wound  in  a  single 
layer  on  a  wooden  cylinder  AB,  as  shown  in  Fig,  100.  is  provided 
with  two  metal  rods  rr  which  are  separated  by  a  small  air  gap. 
One  terminal  of  the  coil  is  connected  to  the  outside  coating  of  a 
Leyden  jar,  and  a  spark  is  allowed  to  jump  from  the  jar  to  the 


Fig.  100. 

other  terminal  of  the  coil  as  shown  in  the  figure.  At  the  instant 
of  formation  of  the  spark  the  total  electromotive  force  between 
the  coatings  of  the  Leyden  jar  begins  to  act  on  the  circuit.  If 
the  coil  consists  of  100  turns  of  wire  wound  on  a  wooden  cylin- 
der 4  centimeters  in  diameter  and  30  centimeters  long,  its  ap- 
proximate inductance  is  0.00005  henry,  so  that,  if  the  electromo- 
tive force  between  the  coatings  of  the  Leyden  jar  is  40,000  volts, 
the  current  begins  to  increase  in  the  coil  at  the  rate  of  800,000,000 
amperes  per  second,  according  to  equation  (49).  The  existence 
of  a  large  electromotive  force  across  the  terminals  of  the  coil 
may  be  shown  by  the  fact  that  the  discharge  of  the  Leyden  jar 

*  For  further  information  concerning  lightning  arresters  see  Franklin  and  Esty, 
Elements  of  Electrical  Engineering,  Vol.  I,  pages  210-219. 


154        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

will  jump  across  the  air  gap  instead  of  going  through  the  coil 
AB.  Thus,  if  the  air  gap  is  one  centimeter  in  length  it  takes 
20,000  volts  to  strike  across  it,  and  if  a  spark  does  strike 
across  this  gap  at  the  instant  of  the  discharge  of  the  Leyden  jar, 
one  may  be  certain  that  the  electromotive  force  between  the  ter- 
minals of  the  coil  was  at  least  20,000  volts  at  the  instant  of  the 
formation  of  the  spark. 

The  protective  action  of  the  choke  coil  in  Fig.  99  depends 
upon  a  rapid  increase  of  current  through  the  coil  during  an  ex- 
tremely short  interval  of  time  just  before  the  gap  g  breaks  down. 
The  dynamo  G  may  be  protected  from  this  very  brief  flow  of 
current  by  connecting  a  condenser  between  the  point  a  and 
earth,  so  that  this  very  brief  flow  of  current  through  the  choke 
coil  need  not  flow  through  the  dynamo,  but  may  go  to  charge  the 
condenser. 

81.  Inductance  of  a  long  solenoid.  —  A  solenoid  is  a  long  coil 
of  wire  ;  two  or  three  layers  of  wire  wound  on  a  long  wooden 
rod,  for  example.  When  the  depth  of  the  winding  of  wire  is 
small  in  comparison  with  the  radius  of  a  solenoid  and  when  the 
length  of  the  solenoid  is  great  in  comparison  with  the  radius,  the 
inductance  of  the  solenoid  in  abhenrys  is  given  by  the  following 

equation 

Z  =  47rW/  (55*) 

in  which  L  is  the  inductance  of  a  solenoid  in  abhenrys,  z  is 
the  number  of  turns  of  wire  on  each  centimeter  of  length  of  the 
solenoid,  r  is  the  radius  of  the  solenoid  in  centimeters,  and  /  is 
the  length  of  the  solenoid  in  centimeters.  The  inductance  of 
the  solenoid  in  henrys  is  given  by  the  equation 


in  which   r  and   /  are  expressed  in  centimeters  as  in  equation 

(55*)- 

Derivation  of  equation  (55).  —  The  intensity  of  the  magnetic 

field  inside  of  the  solenoid  is    H—  qirzl,    according  to  equation 


ELECTRIC   MOMENTUM.     INDUCTANCE.  155 

(34),  where  /  is  the  current  in  the  solenoid  in  abamperes. 
Therefore  the  total  energy  of  the  magnetic  field  inside  of  the 
solenoid  is  equal  to  Trr2  x  /  X  (47r^/)2/87r,  according  to  equation 
(27),  that  is,  the  energy  of  the  magnetic  field  is  given  by  the 
equation 


but  the  energy  of  the  magnetic  field  is  equal  to  \LP,  accord- 
ing to  equation  (48)  so  that  L  is  equal  to  477  Vr2/. 

Equations  ($$a)  and  (55<^)  are  strictly  true  only  for  very  long 
coils  with  thin  windings  of  wire.  These  equations  are  frequently 
useful,  however,  in  determining  the  approximate  inductance  of 
comparatively  short  solenoids  with  thick  windings  of  wire. 

82.  Electric  momentum.  Flux-turns.  —  The  product  of  the  mass  of  a  moving 
body  and  its  velocity  is  called  its  momentum.  The  product  of  the  inductance  of  a 
circuit  and  the  current  is  called  electrical  momentum.  The  term  electrical  momentum 
is  seldom  employed,  the  term  flux-turns  being  more  usual. 

Proposition.  —  The  electrical  momentum  Li  of  a  coil  is  equal  to  the  product  of 
the  flux  through  a  mean  turn  of  the  coil  and  the  number  of  turns  of  wire  in  the  coil, 
that  is, 

Zz  =  Z4>  (56) 

in  which  L  is  the  inductance  of  the  coil  in  abhenrys,  i  is  the  current  in  the  coil  in 
abamperes,  4>  is  the  number  of  lines  of  magnetic  flux  passing  through  a  mean  turn 
of  the  coil,  and  Z  is  the  number  of  turns  of  wire  in  the  coil.  The  truth  of  this 
equation  may  be  made  evident  as  follows  :  The  self-induced  electromotive  force  in  a 
coil  is  due  to  the  increasing  flux  produced  by  the  increasing  current,  so  that  the  self- 
induced  electromotive  force  is  equal  to  —  Z'dbjdt,  according  to  equation  (44), 
where  $  is  the  magnetic  flux  through  a  mean  turn  of  the  coil  due  to  the  current  in 
the  coil.  The  self-induced  electromotive  force  is  also  equal  to  —  L  •  difdt,  accord- 
ing to  equation  (50).  Therefore  we  have 


-whence  by  integrating  *  we  have  Li  =  Z4>. 

83.  The  dependence  of  the  inductance  of  a  coil  on  the  number  of 
turns  of  wire  in  the  coil  and  upon  the  size  of  the  coil.  —  The  de- 

*  This  simple  integration  occurs  so  frequently  in  arguments  of  this  kind  that  it  is 
worth  while  to  consider  its  meaning  as  follows  :  In  order  to  permit  of  a  verbal  ex- 
pression of  equation  (i),  divide  both  members  by  Z,  giving  d^jdt=  Z/ZX  dijdt, 
which  means  that  the  flux  4>  increases  always  LJZ  times  as  fast  as  it  so  that,  if 
4»  and  i  start  from  zero  together,  then  4>  must  always  be  Z/Z  times  as  large  as  i. 


156        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

pendence  of  inductance  upon  the  shape  of  a  coil  is  much  too 
complicated  to  permit  of  its  general  discussion  in  this  text.  The 
ratio  of  the  inductances  of  two  coils  of  exactly  the  same  shape  de- 
pends, however,  in  a  very  simple'*way  upon  the  sizes  of  the  coils 
and  upon  the  relative  number  of  turns  of  wire  in  them,  as  follows  : 

The  inductance  of  a  coil  of  wire  wound  on  a  given  spool  is  pro- 
portional to  the  square  of  the  number  of  turns  of  wire.  —  Thus,  a 
given  spool  wound  full  of  number  16  wire  has  500  turns,  and  an 
inductance  of  0.025  henry;  the  same  spool  wound  full  of  num- 
ber 28  wire  has  ten  times  as  many  turns,  and  its  inductance  is  one 
hundred  times  as  great,  or  2.5  henrys. 

The  inductance  of  a  coil  of  given  shape,  the  number  of  turns  of 
wire  being  unchanged,  is  proportional  to  its  linear  dimensions.  — 
Thus,  if  a  given  spool  of  wire  be  imagined  to  be  increased  in 
dimensions  in  every  detail  in  the  ratio  of  1:10,  size  of  wire  being 
increased  in  the  same  ratio  so  that  the  number  of  turns  will  be 
unchanged,  then  the  inductance  of  the  spool  would  be  increased 
ten  times. 

84.  Kinetic  energy  associated  with  independent  currents  in  two  circuits. 
Definition  of  mutual  inductance.  —  Consider  two  adjacent  circuits  one  of  which 
may  be  called  the  primary  circuit  and  the  other  the  secondary  circuit  to  distinguish 
them.  Let  7j  be  the  current  in  the  primary  circuit  and  72  in  the  secondary  circuit. 
The  total  kinetic  energy  associated  with  these  two  currents  consists  of  three  parts  : 
(a)  A  part  which  is  proportional  to  f:  squared,  (b)  a  part  which  is  proportional  to 

72  squared,  and  (<:)  a  part  which  is  propor- 
tional to    7j/2.     Therefore  we  may  write 

W=  \LJ*  +  £Z272*  4-  MIJi      (i) 

in  which    W   is  the  total  kinetic  energy  of 
the   two  currents,  and    (£7j),    (|72),  and 
M  are   the   proportionality   factors.       The 
quantities    7a    and    Z2    are  the   inductances 
of  the  respective  circuits  inasmuch  as  equa- 
tion (i)  reduces  to  equation  (48)  when  either  current  is  zero.     The  quantity  M  is 
called  the  mutual  inductance  of  the  two  circuits.     It  may  be  either  positive  or  nega- 
tive.    Mutual  inductance  is  expressed  in  terms  of  the  same  units  as  inductance. 

Proof  of  equation  (z).  —  Consider  a  point  p  in  the  neighborhood  of  the  two  cir- 
cuits. Let  7/j,  Fig.  101,  be  the  intensity  at  /  of  the  magnetic  field  due  to  7:  alone, 
and  let  h^  be  the  intensity  at  /  of  the  magnetic  field  due  to  72  alone.  The  result- 
ant magnetic  field  at  /  is  h,  as  shown  in  Fig.  101,  and  we  have 


ELECTRIC   MOMENTUM.     INDUCTANCE.  157 

k*  =  h*  -f  //22  -f  2hjii  cos  0 

Consider  the  energy  A  W  in  a  small  element  of  volume  at  the  point  p.  This 
energy  is  proportional  to  h*  so  that  it  may  be  considered  in  three  parts  which  are 
proportional  to  h*,  to  /#22,  and  to  hjiv  respectively.  But  h^  is  proportional  to 
fv  and  /;2  is  proportional  to  72,  so  that  the  kinetic  energy  in  an  element  of  volume 
at  the  point  /  may  be  considered  in  three  parts  which  are  proportional  respectively 
to  7j2,  to  732,  and  to  I^IV  What  is  true  of  the  energy  in  an  element  of  volume 
at  the  point  /  is  true  of  the  energy  in  every  other  element  of  volume,  that  is,  the 
energy  in  every  element  of  volume  consists  of  three  parts  which  are  proportional 
to  /j2,  to  722,  and  to  7j72,  respectively,  so  that  the  total  energy  consists  of  three 
such  parts. 

PROBLEMS. 

103.  The  current  in  a  circuit  has  a  value  of  26  amperes  at  a 
given  instant.     Three  hundredths  of  a  second  later  the  current  is 
10.3  amperes.     What  is  the  average  rate  of  change  of  the  current 
during  the  interval  ?     Is  this  rate  positive  or  negative  ?     Ans. 
—  523.3  amperes  per  second. 

104.  Calculate  the  kinetic  energy  in  joules  of  a  current  of  160 
amperes  in  a  circuit  having  an  inductance  of  0.05  henry.     Ans. 
640  joules. 

105.  An  electromotive  force  of  25  volts  is  connected  to  a  cir- 
cuit of  which  the  resistance  is  0.6  ohm  and  the  inductance  is  0.05 
henry.     At  what  rate  is  the  current  increasing :  (a)  At  the  instant 
the  electromotive  force  is  connected  to  the  circuit ;   (b)  at  the 
instant  that  the  current  reaches  a  value  of  10  amperes,  and  (c)  at 
the  instant  that  the  current  reaches  a  value  of  3  5  amperes  ?     Ans. 
(a)  500  amperes  per  second,  (fr)  380  amperes  per  second,  (c)  80 
amperes  per  second. 

106.  The  field  winding  of  a  dynamo  has  50  ohms  resistance 
and,  approximately,  7.5   henrys  of  inductance.     Assuming  that 
the  current  grows  in  the  coil  in  accordance  with  equation  (52), 
calculate  the  time   required  for  the  current  in  the  winding  to 
reach  2  amperes  when  the  winding  is  connected  to  a  generator  of 
which  the  electromotive  force  is  no  volts.     Ans.   0.359  second. 

107.  A  current  has  been  left  to  die  away  in  a  circuit  of  0.6 
ohm  resistance  and  0.05  henry  inductance.      Find   the   rate  of 


155        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

change  of  the  current  as  it  passes  the  values  100  amperes,  10 
amperes,  and  one  ampere.  Ans.  —1,200  amperes  per  second, 
—  1 20  amperes  per  second,  —12  amperes  per  second. 

108.  Find  the  approximate  inductance  in  henrys  of  a  cylindrical 
coil  25   centimeters  long,  5   centimeters  mean  diameter,  wound 
with  one  layer  of  wire  containing    150   turns.     Ans.  0.000222 
henry. 

109.  The  choke  coil  of  a  lightning  arrester  consists  of  50  turns 
of  wire  wound  in  one  layer  on  a  cylinder  of  which  the  diameter 
is  1 5  centimeters  and  the  length  is  50  centimeters,     (a)  Calculate 
the  approximate  inductance  of  this  coil,     (ft)  Calculate  the  ap- 
proximate rate  of  increase  of  current  in  the  coil  at  the  instant  that 
a  lightning  discharge  jumps  across  two  centimeters  of  air  in  pref- 
erence to  going  through  the  coil.     Ans.    (#)   o.oooiii   henry. 
(b)  360,000,000  amperes  per  second. 

Note.  —  The  electromotive  force  required  to  strike  across  2  centimeters  of  air  is 
approximately  40,000  volts. 

110.  (a)  Calculate  the  magnetic  flux  through  the  solenoid  which 
is  specified  in  problem  1 1 1  when  a  current  of  30  amperes  flows 
through  it.     (b)  Calculate  the  value  of  the  electrical  momentum 
of  this  current  in  flux-turns,     (a)  4,440  maxwells,     (b)  666,000 
flux-turns. 

111.  The  field  coil  of  a  dynamo  has  5,000  turns  of  wire  and, 
when  a  current  of  one  ampere  flows  through  the  field  winding, 
1,500,000  lines  of  force  are  produced   through  the  field   core. 
Assuming  that  the  flux  is  proportional  to  the  exciting  current, 
find  the  inductance  of  the  field  coil  in  henrys.     Ans.   75  henrys. 

112.  A  battery  having  an  electromotive  force  of  10  volts  and  a 
resistance  of  one  ohm  is  connected  to  a  coil  of  wire  wound  on  an 
iron  core.     The  coil  has  1,000  turns  of  wire  and  its  resistance  is 
4  ohms.     What  is  the  current  in  the  coil  when  the  magnetic  flux 
in  the  core  is  increasing  at  a  rate  of  500,000  lines  per  second  ? 
Ans.  i  ampere. 


ELECTRIC    MOMENTUM.     INDUCTANCE.  159 

113.  A  certain  spool  wound  full  of  wire  o.i  centimeter  in  diam- 
eter has  an  inductance  of  0.08  henry.     The  same  spool  is  wound 
full  of  wire  0.32  centimeter  in  diameter.     What  is  jts  inductance  ? 
Ans.  0.000762  henry. 

114.  A  spool  5  times  as  large  as  the  spool  mentioned  in  prob- 
lem 1 1 6  but  similar  in  shape,  is  wound  with  wire  6  millimeters  in 
diameter.     What  is  its  inductance  ?     Ans.  o.  1 93  henry. 


CHAPTER  VII. 
ELECTRIC   CHARGE.     THE   CONDENSER. 

85.  Electric  charge.  —  A  current  of  water  'through  a  pipe  is  a 
transfer  of  water  along  the  pipe.  Let  q  be  the  amount  of  water 
which,  during  /  seconds,  flows  past  a  given  point  in  the  pipe, 
then  the  quotient  qjt  is  the  rate  of  flow  of  water  through  the 
pipe,  and  this  rate  of  flow  may  be  spoken  of  as  the  strength  / 
of  the  water  current.  Suppose  the  strength  7  of  the  water  cur- 
rent to  be  given  (rate  of  flow  of  water  in  units  of  volume  per 
second)  then  the  amount  of  water  flowing  past  a  given  point  of 
the  pipe  in  t  seconds  is  given  by  the  equation  : 

q-It 

Similarly,  an  electric  current  in  a  wire  may  be  looked  upon  as 
a  transfer  of  electricity  along  the  wire,  and  the  quantity  q  of 
electricity  which  flows  past  a  point  on  the  wire  during  t  seconds 
may  be  defined  as  the  product  of  the  strength  of  the  current  and 

the  time,  that  is, 

q  =  It  (57) 

If  the  strength  of  the  current  is  variable,  then  equation  (57)  must 
be  written  in  the  form 

&q  =  I-bt  (58) 

in  which  A^  is  the  small  quantity  of  electricity  which  flows  past 
a  given  point  on  the  wire  during  the  short  intervals  of  time  A/. 

Quantity  of  electricity  is  usually  spoken  of  as  electric  charge  or 
simply  as  charge. 

Quantity  of  water  is  the  fundamental  and  easily  measured  thing 
in  hydraulics  and  water  current  is  most  conveniently  defined  as 
quantity  of  water  passing  per  second.  In  the  case  of  electricity, 
the  fundamental  and  easily  measured  thing  is  electric  current,  and 

1 60 


ELECTRIC    CHARGE.     THE   CONDENSER.  l6l 

quantity  of  electricity  is  most  conveniently  defined  as  the  product 
of  electric  current  and  time. 

Units  of  electric  charge.  —  The  ampere-second  is  the  amount 
of  electricity  which  flows  in  one  second  through  a  wire  which 
carries  a  current  of  one  ampere.  The  ampere-second  is  usually 
called  the  coulomb.  One  ampere-hour  is  the  quantity  of  elec- 
tricity flowing  in  one  hour  through  a  wire  carrying  one  ampere. 
The  ampere-hour  is  extensively  used  among  electrical  engineers 
in  specifying  the  discharge  capacity  of  storage  batteries.  The 
abcoulomb  is  the  quantity  of  electricity  which  flows  in  one  second 
through  a  wire  carrying  a  current  of  one  abampere.  One  ab- 
coulomb is  equal  to  ten  coulombs. 

86.  Measurement  of  electric  charge.  The  ballistic  galvanom- 
eter.* —  A  very  large  electric  charge  may  be  determined  by  ob- 
serving the  time  during  which  the  charge  will  maintain  a  sensibly 
constant  measured  current.  Thus  a  given  storage  battery  can 
maintain  a  curre'nt,  say,  of  10  amperes  for  8  hours,  so  that  the 
discharge  capacity  of  the  storage  battery  is  equal  to  80  ampere- 
hours.  The  charges  most  frequently  encountered  in  practice, 
however,  are  too  small  to  be  measured  in  this  way,  and  for  such 
charges  the  ballistic  galvanometer  is  used  as  follows  : 

The  charge  to  be  measured  is  sent  through  a  galvanometer  in 
the  form  of  a  pulse  of  electric  current  of  very  short  duration. 
This  pulse  of  current  sets  the  needle  of  the  galvanometer  swing- 
ing. The  maximum  deflection  d  of  the  needle  at  the  first 
swing  is  called  the  throw  of  the  galvanometer,  and  this  throw,  if 
it  is  not  too  large,  is  proportional  to  the  amount  of  charge  q 
which  is  carried  through  the  galvanometer  by  the  pulse  of  cur- 
rent. That  is, 

q  =  kd  (59) 

the  quantity  k  is  called  the  reduction  factor  of  the  galvanometer, 
and  it  is  usually  determined  in  practice  by  observing  the  throw 
produced  by  a  known  charge.  Equation  (59)  is  true  for  a  galva- 

*  See  Chapter  X  for  a  more  complete  discussion  of  the  ballistic  galvanometer  and 
of  the  measurement  of  electric  charge. 
12 


1  62        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

nometer  with  a  heavy  needle  (or  with  a  heavy  moving  coil  in  the 
case  of  the  D'  Arsonval  type  of  instrument)  which  is  not  subject 
to  any  perceptible  air  friction  as  it  vibrates.  Such  a  galvanometer 
is  called  a  ballistic  galvanometer. 

The  reduction  factor  of  a  ballistic  galvanometer  may  be  calcu- 
lated from  the  equation 

„  . 
(60) 


in  which  7  is  a  known  steady  current  which  produces  a  steady 
deflection  b  of  the  galvanometer,  t  is  the  period  of  one  com- 
plete oscillation  of  the  galvanometer  and  X  is  the  ratio  of  two 
successive  throws  of  the  freely  swinging  needle  (or  coil)  of  the 
galvanometer,  swings,  the  so-called  damping  ratio  of  the  galva- 
nometer.* 

87.  The  flow  of  current  in  unclosed  circuits.  Electrically 
charged  bodies.  —  Consider  two  insulated  metal  bodies  A  and 

B,  Fig.  IO2#,  which  at  a  given 
instant  are  connected,as  shown, 
to  the  terminals  of  a  battery  or 
to  any  source  of  electromotive 
force.  When  the  wire  is  con- 
nected a  momentary  pulse  of 
current  flows  through  it  out  of 
one  body  and  into  the  other, 
and  the  bodies  A  and  B  are 
sa^  to  become  charged  with 
electricity. 

The  body  into  which  the  mo- 
Flg'  102a*  mentary  current  flows  is  said  to 

become  positively  charged  and  the  body  out  of  which  the  momen- 
tary current  flows  is  said  to  become  negatively  charged,  that  is, 
the  charge  on  one  body  is  -f  q  and  the  charge  on  the  other  body 

*See  Electrical  Measurements  by  Carhart  and  Patterson,  pages  207-213.  See 
Absolute  Measurements  in  Electricity  and  Magnetism,  by  Andrew  Gray,  Vol.  II. 
pages  390-396.  See  Maxwell's  Electricity  and  Magnetism,  Vol.  II,  pages  374-391, 


ELECTRIC   CHARGE.     THE   CONDENSER.  163 

is  —  q.  Electrically  charged  bodies  always  occur  thus  in  pairs, 
the  positive  charge  on  one  body  being  always  associated  with  an 
equal  negative  charge  on  some  other  body  or  bodies. 

Example.  —  Two  large  sheets  of  tin  foil  separated  from  each 
other  by  waxed  paper  are  connected  through  an  incandescent 
lamp  to  supply  mains.  If  this  arrangement  is  connected  to 
direct-current  supply  mains  a  single  pulse  of  current  flows 
through  the  lamp  at  the  moment  of  connection,  and  the  lamp 
filament  is  not  perceptibly  heated.  If  the  arrangement  is  con- 
nected to  alternating-current  supply  mains  a  pulse  of  current 
flows  through  the  wire  at  every  reversal  of  the  alternating  elec- 
tromotive force  and  the  lamp  filament  may  be  heated  to  incan- 
descence. 

The  electric  field.  The  dielectric.  —  The  region  between  the 
two  bodies  A  and  B,  Fig.  102*2,  is  understood  to  be  filled  with 
some  electrical  insulator  such  as  air,  oil  or  glass.  An  insulator 
between  two  charged  bodies  is  called  a  dielectric.  This  dielectric 
is  the  seat  of  a  peculiar  stress  which  is  called  the  electric  field. 
The  lines  of  force  *  of  this  electric  field  trend  somewhat  as  shown 
in  the  figure,  touching  the  surfaces  of  the  metal  bodies  A  and 
B  at  right  angles.  These  lines  of  force  are  thought  of  as  going 
out  from  the  positively  charged  body  and  coming  in  towards  the 
negatively  charged  body. 

Electrostatic  attraction.  —  The  charged  bodies  A  and  B,  Fig. 
IO2#,  attract  each  other.  This  attraction,  which  is  called  elec- 
trostatic attraction,  shows  that  the  lines  of  force  of  an  electric 
field  are  in  a  state  of  tension  and  have  a  tendency  to  shorten  very 
much  as  the  lines  of  force  'in  a  magnetic  field.  This  tension  of 
the  lines  of  force  pulls  outwards  on  the  surface  of  A  and  on  the 
surface  of  B  at  each  point. 

The  electrostatic  attraction  of  two  metal  bodies  which  are  con- 
nected to  a  battery  or  dynamo  may  be  shown  as  follows  :  A  gold 
leaf  is  hung  alongside  of  a  vertical  brass  strip.  When  the  gold 

*  The  electric  field  is  similar  in  many  respects  to  the  magnetic  field,  having  a  defi- 
nite intensity  and  a  definite  direction  at  each  point. 


164        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

leaf  is  connected  to  one  terminal  of  a  dynamo  and  the  brass  strip 
to  the  other  terminal,  the  gold  leaf  is  attracted  by  the  brass.  It 
is  necessary  in  this  arrangement  to  cover  the  face  of  the  brass 
strip  with  a  layer  of  paper  to  avoid  short-circuiting  the  dynamo 
through  the  gold  leaf. 

The  outward  pull  of  the  electric  field  on  the  surface  of  a 
charged  body  is  very  strikingly  shown  by  pouring  a  viscous 
liquid  over  the  sharp  lip  of  a  charged  metal  ladle.  The  liquid  is 
pulled  into  fine  jets  by  the  lines  of  force  which  emanate  from  the 
surface  of  the  liquid  as  it  passes  over  the  lip.  When  melted  rosin 
is  used  in  this  way  the  jets  congeal  into  very  fine  fibers  which 
float  about  in  the  air. 

Need  of  high  electromotive  force  and  of  good  insulation.  —  The 
phenomena  described  above,  in  fact  most  of  the  phenomena  of 
electrostatics,  are  easily  perceptible  only  when  the  bodies  are 
charged  by  electromotive  forces  of  many  thousands  of  volts.  The 
most  convenient  method  of  producing  these  large  electromotive 
forces  is  by  means  of  the  Holtz  or  Wimshurst  electrical  machine, 
and,  when  such  a  machine  is  used,  the  bodies  A  and  B  must  be 
well  insulated,  because  such  electrical  machines  cannot  supply 
charge  at  a  rapid  rate,  that  is,  such  machines  can  deliver  only 
very  small  currents.  (See  Arts.  106  and  108.) 

The  electric  spark.  —  When  the  electromotive  force  acting  to 
charge  two  bodies  A  and  By  Fig.  IO2#,  is  increased  more  and 
more,  a  value  is  eventually  reached  which  breaks  down  or  ruptures 
the  dielectric  and  allows  the  charge  on  the  bodies  to  pass  in  the 
form  of  an  electric  spark. 

Mechanical  analogue  of  electrically-charged  bodies  and  of  the 
electric  field. — Imagine  two  cavities  A  and  B,  Fig.  102$,  in  an 
extended  elastic  solid  such  as  rubber  or  jelly.  Imagine  these 
cavities  to  be  filled  with  water  and  to  be  connected  to  a  pump  by 
means  of  a  pipe  so  that  the  pump  may  draw  a  certain  amount  of 
water  out  of  one  cavity  and  force  it  into  the  other,  thus  causing 
one  cavity  to  contract  and  the  other  cavity  to  expand,  and  causing 
the  surrounding  mass  of  rubber  or  jelly  to  be  strained,  the  lines 


ELECTRIC    CHARGE.     THE   CONDENSER. 


I6S 


of  stress  or  strain  being  somewhat  as  shown  in  the  figure.  The 
expanded  cavity  is  analogous  to  a  positively  charged  body,  the 
contracted  cavity  is  analogous  to  a  negatively  charged  body,  the 
stressed  condition  of  the  rubber  or  jelly  is  analogous  to  the  elec- 
tric field  between  two  charged  bodies  and  the  pressure-difference 
of  the  pump  is  analogous  to 
the  electromotive  force  of 
the  battery  in  Fig.  iO2a. 

88.  Electrostatic  capac- 
ity. The  condenser. — The 
amount  of  charge  q  which 
flows  out  of  B  and  into  A, 
Fig.  I  O2a,  when  the  battery 
is  connected  is  proportional 
to  the  electromotive  force 
of  the  battery.  Therefore 
we  may  write 

9  —   CE  (6l)  Fig.  102b. 

in  which  q  is  the  charge  that  is  drawn  out  of  B  and  forced  into 
A,  in  Fig.  iO2a,  by  a  battery  of  which  the  electromotive  force  is 
E,  and  C  is  a  constant  depending  upon  the  size  and  shape  of 
A  and  B  and  upon  the  nature  of  the  intervening  dielectric. 
This  quantity  C  is  called  the  electrostatic  capacity  or  simply  the 
capacity  of  the  pair  of  bodies  A  and  B.  If  the  bodies  A  and 
B  are  in  the  form  of  flat  plates  of  metal  separated  by  a  thin  layer 
of  dielectric  their  electrostatic  capacity  is  large.  Such  an  arrange- 
ment is  called  a  condenser.  Condensers  are  usually  made  of 
sheets  of  tin  foil  separated  by  sheets  of  waxed  paper  or  mica. 
The  Ley  den  jar  is  a  condenser  made  by  coating  the  inside  and 
outside  of  a  glass  jar  with  tin  foil. 

Measurement  of  capacity.  —  The  simplest  method  of  measuring 
the  capacity  of  a  condenser  is  to  charge  the  condenser  by  a  bat- 
tery of  known  electromotive  force  E  and  then  measure  the 


1 66        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

charge  q(=  CE)  by  discharging  the  condenser  through  a  bal- 
listic galvanometer.  See  Chapter  X. 

Units  of  capacity.  — A  condenser  is  said  to  have  a  capacity  of 
QV^  farad  when  one  coulomb  of  charge  is  drawn  out  of  one  plate 
and  forced  into  the  other  plate  by  an  electromotive  force  of  one 
volt ;  C  in  equation  (6 1 )  is  expressed  in  farads  when  q  is 
expressed  in  coulombs  and  E  in  volts.  The  farad  is  an 
extremely  large  capacity  as  compared  with  capacities  ordinarily 
met  with  in  practice,  and  the  microfarad  (one  millionth  of  a  farad) 
is  frequently  used  as  a  unit.  The  term  abfarad  is  occasionally 
used  to  designate  the  c.g.s.  unit  of  capacity.  A  condenser  would 
have  one  abfarad  of  capacity  if  one  abcoulomb  of  charge  would 
be  drawn  out  of  one  plate  and  forced  into  the  other  plate  by  an 
electromotive  force  of  one  abvolt.  One  abfarad  is  equal  to 
io9  farads. 

Electric  absorption. — When  a  condenser,  which  has  been 
charged  for  some  time,  is  discharged  and  then  left  standing,  a 
small  amount  of  additional  charge  collects  on  the  condenser 
plates  so  that  a  second  or  third  discharge  may  be  taken  from  the 
condenser.  It  seems  as  if  a  portion  of  the  initial  charge  on  the 
condenser  were  absorbed  by  the  dielectric,  this  absorbed  charge 
being  slowly  given  back  to  the  condenser  plates  when  these  have 
been  discharged.  This  phenomenon  of  electric  absorption  is 
strictly  analogous  to  the  following:  A  rubber  tube  which  is 
stretched  for  some  time  and  then  released,  comes  nearly  back  to 
its  initial  length  at  once,  and  then  continues  to  shorten  for  a  long 
time.  If  the  end  of  the  tube  is  fixed  immediately  after  the 
release,  the  tendency  of  the  tube  to  continue  to  shorten  will 
develop  a  stretched  condition  in  the  tube  which  will  show  itself 
by  a  sudden  slight  shortening  when  the  tube  is  released  a 
second  time. 

89.  Mechanical  analogue  of  the  condenser.  —  Figure  103  shows 
two  metal  plates  AA  and  BB  separated  by  a  dielectric  DD 
and  connected  to  a  battery.  Figure  104  shows  a  box  separated 
into  two  compartments  AA  and  BB  by  means  of  a  rubber 


ELECTRIC    CHARGE.     THE   CONDENSER. 


I67 


diaphragm  DD  and  the  two  compartments  are  connected  to  a 
pump  P.  The  electromotive  force  of  the  battery  in  Fig.  103 
forces  a  certain  amount  of  charge  q  into  the  plate  AA,  draws 
the  same  amount  of  charge  out  of  the  plate  BB,  and  subjects 
the  dielectric  DD  to  an  electrical  stress.  The  pressure-differ- 
ence developed  by  the  pump  P  in  Fig.  104  forces  a  certain 


B 


wire 


U'l'l'l' 


jpire- 

-M 

Fig.  103. 

amount  of  water  q  into  the  compartment  AA,  draws  the  same 
amount  of  water  out  of  the  compartment  BB,  and  subjects  the 
rubber  diaphragm  DD  to  a  mechanical  stress.  If  the  pump  P 
in  Fig.  104  is  removed  and  the  two  compartments  connected  by 
a  pipe,  the  mechanical  stress  of  the  diaphragm  DD  will  be 
relieved  by  a  momentary  flow  of  water  from  A  to  B  through 
the  pipe.  If  the  battery  in  Fig.  103  is  removed  and  the  two 
plates  connected  by  a  wire,  the  electrical  stress  of  the  dielectric 
DD  will  be  relieved  by  a  momentary  flow  of  electric  current 
through  the  wire.  When  the  pump  P  in  Fig.  104  is  connected, 
as  shown,  it  causes  water  to  flow  out  of  B  into  A  until  the 
pressure-difference  developed  by  the  pump  is  balanced  by  the 
elastic  reaction  of  the  diaphragm  DD,  and  the  amount  of  water 
drawn  out  of  B  and  forced  into  A  is  proportional  to  the 
pressure-difference  developed  by  the  pump.  When  the  battery 
B  is  connected  to  the  plates  in  Fig.  103,  it  causes  an  electric 
current  to  flow  out  of  B  and  into  A  until  the  electromotive 


1 68        ELEMENTS  OF  EEECTRICITY  AND  MAGNETISM. 

force  of  the  battery  is  balanced  by  what  one  may  perhaps  call  the 
electro-elastic  reaction  of  the  dielectric  DD,  and  the  amount  of 
charge  drawn  out  of  B  and  forced  into  A  is  proportional  to 
the  electromotive  force  of  the  battery. 

The  extent  to  which  the  diaphragm  DD,  Fig.  104,  yields  is 
measured  by  the  amount  of  water  q  which  is  drawn  out  of  B 
and  forced  into  A,  and  the  yield  per  unit  of  Pressure-difference  is 
a  sort  of  coefficient  of  elasticity  of  the  diaphragm.  The  extent 
to  which  the  dielectric  DD,  Fig.  103,  yields  is  measured  by  the 
amount  of  charge  q  which  is  drawn  out  of  the  plate  B  and 
forced  into  plate  A,  and  the  amount  of  yield  per  unit  of  electro- 
motive force  of  the  battery  (qr/J£=C)  is  a  sort  of  coefficient  of  elec- 
tro-elasticity of  the  layer  of  dielectric  and  it  is  called  the  capacity 
of  the  condenser.  It  is  important  to  remember  that  the  capacity 
of  a  condenser  is  not  analogous  to  the  cubic  capacity  of  a  vessel 
but  that  it  is  analogous  to  the  cubic  capacity  of  a  rubber  bag,  the 
amount  of  water  that  a  rubber  bag  will  hold  depends  upon  the 
pressure. 

90.  Inductivity  *  of  a  dielectric.  —  If  the  diaphragm  DD  in 
Fig.  104  were  made  of  a  stiff  material  like  steel,  the  amount  of 
yield  per  unit  of  pressure-difference  would  be  very  much  less 
than  if  the  diaphragm  were  made  of  a  substance  like  rubber. 
This  is  analogous  to  the  fact  that  the  capacity  of  the  condenser 
AB,  Fig.  103,  with  plates  of  a  given  size  at  a  given  distance 
apart  depends  upon  the  nature  of  the  dielectric  between  the  plates. 
The  quotient :  Capacity  of  a  condenser  with  given  dielectric, 
divided  by  the  capacity  of  the  same  condenser  with  air  between  its 
plates  is  called  the  inductivity  of  the  dielectric.  For  example, 
the  inductivity  of  petroleum  is  about  2.04,  that  is,  the  capacity 
of  a  given  condenser  is  about  2.04  times  as  great  when  the  dielec- 
tric is  petroleum  as  it  is  when  the  dielectric  is  air.  A  condenser 
is  called  an  air  condenser,  a  mica  condenser,  a  paraffin  condenser, 
etc.,  according  to  the  dielectric  between  its  plates.  The  accom- 
panying table  gives  the  inductivities  of  a  few  dielectrics. 

*  Sometimes  called  specific  inductive  capacity. 


ELECTRIC   CHARGE.      THE   CONDENSER.  169 

TABLE. 

Inductivities  of  various  substances. 


Glass 3-10 

Sulphur 2.24-3.84 

Vulcanite 2.50 

Paraffin 1.68-2.30 

Rosin 1.77 

Wax  .  ,     1.86 


Shellac 2.95-3.60 

Mica 4-8 

Quartz 4.5 

Turpentine 2.15-2.43 

Petroleum 2.04-2.42 

Water 73~9O 


The  inductivity  of  a  Dielectric  is  determined  by  measuring  the 
capacity  of  a  condenser  first  with  air  between  its  plates  and  then 
with  the  given  dielectric  between  its  plates.  See  Chapter  X. 

91.  Dependence  of  the  capacity  of  a  condenser  upon  size  and 
distance  apart  of  its  plates.  —  Using  a  ballistic  galvanometer  as 
explained  in  Art.  88,  it  may  be  shown  experimentally*  that  the 
capacity  C  of  a  condenser,  an  air  condenser,  for  example,  is 
proportional  f  to  the  area  a  of  one  of  its  plates,  and  inversely 
proportional  to  the  distance  x  between  its  plates  ;  that  is,  C  is 
proportional  to  ajx,  so  that  we  may  write 


in  which  C  is  the  capacity  of  the  air  condenser,  a  is  the  area 
of  one  plate  (sectional  area  of  the  dielectric),  x  is  the  distance 
between  the  plates,  and  I  IB  is  a  constant. 

When  C  is  expressed  in  farads,  a  in  square  centimeters,  and 
x  in  centimeters,  then  the  value  of  I/  B,  as  determined  by 
experiment  is  884  x  io~16,  so  that  equation  (62)  becomes 

Garads  =  884  X     IO~W  X    ^  (63) 

3C 

*  Indeed  it  may  be  shown  from  geometrical  considerations  that  C  must  be  pro- 
portional to  a\x  ;  the  value  of  the  proportionality  factor  must  however  be  determined 
by  observation.  It  is  possible  to  calculate  the  value  of  l\B  from  the  observed 
velocity  of  light  as  explained  in  Art.  146. 

f  When  a  is  large  compared  with  x,  the  non-uniformity  of  the  electric  field  near 
the  edges  of  two  parallel  oppositely  charged  plates  is  negligible,  and  it  is  ignored 
throughout  this  discussion. 


I/O        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

in  which  k  is  the  inductivity  of  the  dielectric,  x  is  the  thickness 
of  the  dielectric  in  centimeters,  and  a  is  the  area  in  square 
centimeters  of  one  plate  of  the  condenser  (sectional  area  of  the 
dielectric). 

92.  Work  done  by  an  electromotive  force  in  pushing  a  given 
amount  of  charge  through  a  circuit.  —  Consider  an  electromotive 
force    E    maintaining  a  current    /    in   a   circuit.     The  rate  at 
which  this  electromotive  force  does  work  is  equal  to    El,    which, 
multiplied  by  a  time   /,    gives  the  work  done  during  that  time, 
so  that    W  —  Elt.     But  the  product   //  is  equal  to  the  charge  q 
which  is  transferred  during  the  time    t,    therefore  we  have 

W=Eq  (64) 

in  which  W  is  the  work  done  by  an  electromotive  force  E  dur- 
ing the  time  that  charge  q  is  pushed  through  the  circuit.  The 
work  W  is  expressed  in  joules  when  E  is  expressed  in  volts 
and  q  in  coulombs. 

93.  The  potential  energy  of  a  charged  condenser.  —  A  charged 
condenser  represents  a  store  of  potential  energy  in  much  the  same 
way  that  the  distorted  diaphragm    DD   in  Fig.  104  represents  a 
store  of  potential  energy,  or  in  the  same  way  that  a  bent  spring 
represents  a  store  of  potential  energy.     When  a  spring  is  bent, 
the  bending  force  is  at  first  equal  to  zero,  it  increases  in  propor- 
tion to  the  amount  of  bending,  and  the  average  value  of  the 
bending  force  is  equal  to  one  half  its  ultimate  value  (that  is,  the 
value  which  corresponds  to  a  given  amount  of  bend).     Let   E 
be  the  ultimate  value  of  the  bending  force  and    q   the  distance 
through  which  the  end  of  the  spring  is  moved,  then    \E  is  the 
average  value  of  the  bending  force,  which,  multiplied  by  q,   gives 
the  work  done  in  bending  the  spring  or  the  potential  energy  of  the 
bent  spring.     Therefore,  the  potential  energy  of  the  bent  spring 

is  given  by  the  equation 

W=\Eq 

in  which  E  is  the  ultimate  value  of  the  bending  force,  and  q  is 


ELECTRIC   CHARGE.      THE   CONDENSER. 


171 


the  distance  through  which  the  end  of  the  spring  is  moved  dur- 
ing the  bending. 

In  a  similar  manner,  it  may  be  shown  that  the  potential  energy 

of  a  charged  condenser  is 

(65*) 


in  which  E  is  the  electromotive  force  between  the  plates  of  the 
charged  condenser,  and  q  u  t'.c  amount  of  charge  which  has 
been  drawn  out  of  one  plate  and  pushed  into  the  other.  The 
potential  energy  W  is  expressed  in  joules  when  E  is  expressed 
in  volts  and  q  in  coulombs. 

A  weight  is  hooked  to  the  lower  end  of  a  vertical  spring,  as 
shown  in  Fig.  105,  and  then  the  weight  is  released.  In  this  case 
the  full  value  of  the  weight  acts 
upon  the  spring  from  the  start, 
and  the  weight  oscillates  up 
and  down  for  some  time  before 
coming  to  rest  in  its  equilibrium 
position.  Let  E  be  the  pull 
of  the  earth  upon  the  weight 
and  let  q  be  the  distance  from 
the  initial  to  the  equilibrium 
position,  then  the  total  work 
done  by  gravity  after  the  weight 
has  come  to  rest  is  equal  to  Eq 
(pull  of  earth  multiplied  by  dis- 


fcinitial  position 


------  ^equilibrium  positi 


tance  through  which  the  weight 
has  moved),  but  the  potential 
energy  which  is  stored  in  the 
spring  after  the  weight  comes 
to  rest  in  its  equilibrium  posi- 
tion is  equal  to  \Eq  that  is, 
one  half  of  the  work  which  has  been  done  on  the  weight  by 
gravity  is  stored  in  the  spring  as  potential  energy  and  the  re- 
mainder of  the  work  has  been  dissipated  by  the  oscillations  of 
the  weight. 


jl  _  ^extreme  position 

6- 

Fig.  105. 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

When  a  battery  is  connected  to  the  terminals  of  a  condenser 
the  full  electromotive  force  of  the  battery  begins  to  act  at  once, 
and  the  current  surges  back  and  forth  through  the  circuit  until 
the  system  finally  settles  to  equilibrium.  When  this  final  state 
of  equilibrium  is  reached,  a  definite  amount  of  charge  q  will 
have  been  pushed  into  the  condenser  by  the  battery,  and  the 
total  amount  of  work  done  by  the  battery  will  be  Eq  ;  but  the 
amount  of  potential  energy  stored  in  the  condenser  is  \Eq,  and 
therefore  an  amount  of  work  \Eq  has  been  dissipated  by  the 
electrical  oscillations  of  the  system,  exactly  as  in  the  case  of  the 
spring  and  weight  above  described. 

In  order  that  all  the  work  done  in  stretching  a  spring  may  be 
stored  in  the  spring  as  potential  energy,  the  stretching  force  must 
begin  at  zero  and  increase  gradually  as  the  spring  is  bent  more 
and  more  ;  in  order  that  all  the  work  done  in  charging  a  con- 
denser may  be  stored  in  the  condenser  as  potential  energy,  the 
charging  electromotive  force  must  begin  at  zero  and  increase 
gradually  as  the  condenser  becomes  charged.  If  the  final  value 
of  the  charging  electromotive  force  is  E  its  average  value  is 
\E,  which,  multiplied  by  the  amount  of  charge  q  that  has  been 
pushed  into  the  condenser,  gives  the  potential  energy  of  the 
condenser. 

The  potential  energy  of  a  charged  condenser  may  be  expressed 
in  terms  of  E  and  q,  or  in  terms  of  C  and  E,  or  in  terms 
of  C  and  q  by  using  equation  (61).  Thus,  by  substituting 
CE  for  q,  equation  (6  5*2)  becomes 

W= 


and  by  substituting   qjC  for   E,    equation  (650)  becomes 


94.  Transference  of  charge  by  a  moving  ball.  Intensity  of  elec- 
tric field.  —  Two  metal  plates  A  and  B,  Fig.  106,  are  con- 
nected to  a  battery  of  which  the  electromotive  force  is  E,  and  a 


ELECTRIC    CHARGE.      THE   CONDENSER. 


'73 


silk 
thread 


very  small  metal  ball  b  is  suspended  between  A  and  B  by  a 
silk  thread.  If  this  ball  is  started  it  continues  to  vibrate  back  and 
forth  from  plate  to  plate,  and  at  each  movement  it  carries  across 
a  definite  amount  of  charge  q.  Every 
time  the  ball  carries  charge  q  across 
from  plate  to  plate  an  amount  of 
charge  q  flows  through  the  battery, 
the  battery  does  an  amount  of  work 
equal  to  Eq  and  this  work  reappears 
as  mechanical  work  done  on  the  ball 
as  it  is  pushed  across  from  plate  to 
plate  by  the  electric  attraction  or  re- 
pulsion. Let  F  be  the  force  which 

pushes    on  the  ball,  then  Fx   is  the     ^ |.|.|.| 

work   done  by  this  force  in  pushing 
the  ball  across  from  plate  to  plate,* 

_  air n        so  that   Fx  =  Eg,    or 


wire 


•*•? 


air  — 


air 


B 


-xq 


(0 


Fig.  107. 


Any  region  in  which  a  charged  body  is 
acted  upon  by  a  force  f  is  called  an  electric 
field ;  thus  the  region  between  the  plates 
A  and  B,  Fig.  106,  is  an  electric  field,  as 
indicated  by  the  fine  lines  of  force  in  Fig. 
107. 

The  force  F  with  which  an  electric  field 
pulls  on  a  charged  body  (of  small  size) 
placed  at  a  given  point  in  the  field  is  pro- 
portional to  the  charge  q  on  the  body  so 
that 


*  The  ball  is  supposed  to  be  quite  small  so  that  the  distance  moved  by  it  may  be 
taken  to  be  equal  to  the  distance  x  between  the  plates.  Under  these  conditions  the 
force  which  acts  upon  the  ball  is  constant  throughout  its  movement  from  plate  to  plate. 

f  That  is,  a  force  which  depends  upon  the  charge  on  the  body  and  which  does  not 
exist  when  the  body  has  no  charger 


1 74        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


F=fq  (66) 

in  which   /  is  the  proportionality  factor.     This  quantity  f   is 
called  the  intensity  of  the  electric  field  at  the  point. 

Comparing  equations  (i)  and  (66),  it  is  evident  that  the  intensity 
of  the  electric  field  between  the  parallel  plates    AB   in  Fig.  106  is 


/-f 


(67) 


That  is,  the  intensity  of  the  electric  field  between  the  plates  is 
equal  to  the  electromotive  force  between  the  plates  divided  by 
the  distance  between  the  plates.  When  E  is  expressed  in  volts 
and  x  in  centimeters,  the  field  intensity  is  expressed  in  volts  per 
centimeter.  The  intensity  of  an  electric  field  may  also  be  expressed 
in  abvolts  per  centimeter. 

Direction  of  electric  field  at  a  point.  —  The  direction  of  an  elec- 
tric field  at  a  point  is  the  direction  in  which  the  field  pulls  on  a 
positively  charged  body  placed  at  that  point.  A  line  of  force  in 
an  electric  field  is  a  line  drawn  so  as  to  be  in  the  direction  of  the 
field  at  each  point. 

95.  Dielectric  strength. — The  ability  of  a  dielectric  to  with- 
stand electrical  stress  or  electric  field  is  called  the  strength  of  the 
dielectric.  The  strength  of  a  dielectric  is  measured  by  the  inten- 
sity of  the  electric  field  in  volts  per  centimeter  which  is  just  sufft- 


Fig.  108. 


Fig.  109. 


ELECTRIC    CHARGE.     THE    CONDENSER.  175 

cient  to  rupture  it.  Thus,  air  at  ordinary  atmospheric  pressure 
is  ruptured  by  an  electric  field  of  which  the  intensity  is  about 
24,000  volts  per  centimeter,  and  kerosene  is  ruptured  by  an  elec- 
tric field  of  which  the  intensity  is  about  50,000  volts  per  centi- 
meter. The  dielectric  strength  of  a  substance  varies  greatly  with 
its  degree  of  purity. 

When  an  insulating  substance  is  placed  between  two  flat  metal 
plates  A  and  B,  as  shown  in  Fig.  108,  the  substance  is  sub- 
jected to  a  uniform  electrical  stress  (uniform  electric  field)  when 
the  plates  are  connected  to  an  electrical  machine,  and  the  electro- 
motive force  required  to  rupture  the  substance  is  quite  accurately* 
proportional  to  the  thickness  of  the  insulating  layer,  provided  the 
insulating  substance  is  homogeneous  like  air  or  oil ;  or,  in  other 
words,  a  fairly  definite  intensity  of  electric  field  (volts  per  centi- 
meter) is  required  to  rupture  a  homogeneous  substance  like  air  or 
oil,  and  such  a  substance  has  therefore  a  fairly  definite  dielectric 
strength.  Most  solid  substances,  however,  are  non-homogene- 
ous. Thus,  the  rubber  gum  which  is  extensively  used  for  insu- 
lating wires  is  "  filled  "  with  finely  divided  clay  and  is  therefore 
non-homogeneous.  Sheets  of  window  glass  are  usually  filled 
with  fine  bubbles  and  are  therefore  non-homogeneous.  Thick 
sheets  of  vulcanized  fiber  are  usually  charged  with  moisture  in 
the  interior  and  dry  near  the  surface,  and  they  are,  therefore,  non- 
homogeneous.  The  electromotive  force  required  to  rupture  a 
non-homogeneous  substance  is  not  even  approximately  propor- 
tional to  the  thickness  of  the  layer,  and  it  is  therefore  customary 
to  specify  the  dielectric  strength  of  solid  insulating  substances  by 
giving  the  electromotive  force  required  to  rupture  a  specified 
thickness. 

The  least  roughness  of  the  surface  of  the  metal  plates  A  and 
B,  in  Fig.  108,  or  particles  of  dust  floating  in  the -dielectric,  pro- 
duce great  variations  in  the  value  of  the  electromotive  force 
required  to  rupture  a  dielectric.  The  action  of  these  irregular- 
ities of  surface  and  of  floating  particles  is  shown  somewhat  exag- 

*See  Art.  127. 


1/6        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


gerated  in  Fig.  109.  In  this  figure  a  floating  particle  and  a 
minute  projecting  point  on  the  plate  B  are  represented.  The 
intensity  of  the  electric  field  near  the  point  and  near  the  ends  of 
the  particle  is  much  greater  -'than  the  average  intensity  Ejx 
between  the  plates,  and  the  dielectric  begins  to  give  way  at  these 

places  when  the  field  intensity  there  reaches  the  breaking  value. 

«. 

TABLE.* 

Dielectric  strengths. 


Substance. 

Strength  in 
Volts  per  Cen- 
timeter. 

Substance. 

Strength  in 
Volts  per  Cen- 
timeter. 

Oil  of  turpentine 

94,000 

Beeswaxed  paper 

54O,OOO 

Paraffine  oil 

87,000 

Air  (thickness  5  cm.  ) 

23,800 

Olive  oil 

82,000 

C02 

22,700 

Paraffine  (melted) 

56,000 

0 

22,'2OO 

Kerosene  oil 

50,000 

H 

I5,IOO 

Paraffine  (solid) 

130,000 

Coal  gas          " 

22,3OO 

Paraffined  paper 

360,000 

Lines  of  force  of  the  electric   field   between   two  oppositely 
charged  metal  spheres  are  shown  in  Fig.  no.     In  this  case  the 

electric  field  is  not  uniform, 
and  the  intensity  of  the  elec- 
tric field  in  volts  per  centi- 
meter near  the  surface  of  one 
of  the  spheres  may  be  suffi- 
cient to  start  a  rupture,  al- 
though the  intensity  of  the 
field  at  a  distance  from  the 
surface  of  one  of  the  spheres  may  be  much  less  than  that  which 
corresponds  to  the  rupture  of  the  dielectric.  Furthermore,  the 
electric  field  intensity  in  Fig.  no  is  not,  of  course,  equal  to  the 
electromotive  force  between  the  spheres  divided  by  their  distance 
apart,  because  the  field  is  non-uniform.  Therefore  the  electro- 
motive force  required  to  rupture  a  dielectric  between  two  metal 
spheres  is  not  proportional  to  their  distance  apart. 

*From  the  measurements  of  Macfarlane  and  Pierce,   Physical  Review,  Vol.  I, 
page  165. 


Fig.  110. 


ELECTRIC   CHARGE.     THE   CONDENSER. 


177 


96.  The  spark  gauge.  —  The  electromotive  force  required  to 
produce  a  spark  between  two  polished  metal  spheres  of  given  size 
in  air  varies  in  a  definite  manner  with  the  length  of  the  air  gap. 
If  the  electromotive  forces  required  for  different  distances  be  once 
determined  by  observation,  then  any  electromotive  force  may  be 
determined  by  measuring-  its  sparking  distance  between  the  pair 
of  spheres.  An  arrangement  for  measuring  electromotive  force 
in  this  way  is  called  a  spark  gauge  or  a  spark  micrometer.  The 
spark  gauge  is  adapted  only  to  high  electromotive  forces,  and  the 
results  obtained  by  it  are  subject  to  large  errors. 

TABLE. 

Sparking  distances  in  air  at  18°  C.  and  745  mm.* pressure, 
s  =  length  of  air  gap  in  centimeters. 
r  =  radius  of  spheres  in  centimeters. 


J 

r  =  0.25 

r  =  0.5 

r=  i.o 

r  =  2.s  cm. 

cm. 

Volts. 

Volts. 

Volts. 

Volts. 

O.I 

4,830 

4,800 

4,710 

0.2 

8,370 

8,370 

8,100 

o.3 

",370 

H,370 

n,37o 

0.4 

13,800 

14,400 

14,400 

0-5 

15,600 

17,400 

17,400 

18,300 

0.6 

17,100 

19,800 

20,400 

21,600 

o.7 

18,300 

21,900 

23,100 

24,600 

0.8 

18,900 

24,000 

26,100 

27,300 

0.9 

19,500 

25,500 

28,800 

30,000 

i.o 

20,  100 

27,000 

31,200 

32,700 

i.i 

20,700 

33,3oo 

35,700 

1.2 

21,000 

35,4oo 

38,400 

1-3 

21,600 

37,200 

4I,IOO 

1.4 

21,900 

38,700 

43,800 

i.S 

22,200 

40,  200 

46,200 

1.6 

41,400 

48,600 

97.  Electric  flux.  —  The  product  of  the  intensity  of  an  electric 
field  and  an  area  at  right  angles  to  the  direction  of  the  field  is 
called  the  electric  flux  across  the  area ,  that  is,  we  may  write 

3>  =fa  (68) 

in  which   <I>   is  the  electric  flux  across  a  square  centimeters  of 

*  Heydweiller,  Wied.  Ann.  48,  p.  235,  1893. 
13 


178         ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

area  at  right  angles  to  an  electric  field  of  intensity  /.  When  / 
is  expressed  in  volts  per  centimeter  and  a  in  square  centimeters, 
the  flux  O  is  expressed  in  terms  of  a  unit  which  may  be  called 
the  volt-centimeter.  Electric  ffux  is  in  many  respects  similar  to 
magnetic  flux,  but  the  two  must  not  be  confused,  although  the 
same  letter  4>  is  here  used  for  both. 

\. 

98  Amount  of  electric  flux  which  emanates  from  an  electric 
charge.  —  It  was  shown  in  Art.  39  that  ^irm  units  of  magnetic 
flux  emanate  from  a  magnet  pole  of  which  the  strength  is  m, 
that  is,  the  strength  of  a  magnet  pole  may  be  expressed  in  terms 
of  the  magnetic  flux  which  emanates  from  it.  There  is  also  a 
simple  proportional  relationship  between  the  amount  of  charge  on 
a  body  and  the  amount  of  electric  flux  which  emanates  from  the 
body,  and  the  amount  of  charge  on  a  body  may  be  expressed  in 
terms  of  the  electric  flux  which  emanates  from  it.  The  relation- 
ship between  electric  charge  and  flux  will  be  established  for  the 
simplest  case,  namely,  the  case  in  which  a  charge  is  spread  uni- 
formly over  a  flat  surface,  as  on  the  flat  metal  plate  of  a 
condenser. 

Consider  the  two  parallel  metal  plates  AA  and  BB,  Fig. 
107,  the  area  of  each  plate  being  a  square  centimeters,  their  dis- 
tance apart  being  x  centimeters,  and  the  electromotive  force  be- 
tween them  being  E  volts.  The  lines  of  force  of  the  electric 
field  in  the  region  between  the  plates  are  indicated  by  the  fine 
arrows,  and  the  intensity  of  this  field  is  equal  to  Ejx,  according 
to  equation  (67),  so  that  the  total  electric  flux  4>  emanating  from 
the  plate  AA  is  a  x  Ejx.  The  capacity  of  the  condenser  AA 
BB  in  farads  is  884  x  io~16  x  ajx,  according  to  equation  (63), 
so  that  the  charge  on  one  plate  is  equal  to  884  x  io~16  x  aEjx 
(q  ==  CE)  which  is  equal  to  884  x  icr16  x  <£,  because  aEjx  is 
equal  to  4> ;  therefore 

2=884x  io~16x  <£ 
or 

®  =  Bq  (69) 

in  which 


ELECTRIC    CHARGE.     THE   CONDENSER 


179 


B=  i  131  x  io13  (70) 

(see  Art.  91),  that  is,  B  lines  of  electric  flux  emanate  from  one 
coulomb  of  positive  charge  or  converge  towards  one  coulomb 
of  negative  charge.  It  is  a  great  help  towards  a  clear  under- 
standing of  electrostatics  to  think  of  electric  charge,  positive  or 
negative -,  as  the  beginning  or  ending  of  lines  of  electric  force. 

99.  Electric  field  due  to  a  concentrated  charge.  —  Consider  a 
concentrated  positive  charge  q  from  which  lines  of  electric  force 
emanate  in  all  directions,  as  shown  in  Fig.  1 1 1,  and  let  it  be  re- 


Fig.  111. 

quired  to  find  the  intensity  /  of  the  electric  field  at  a  point  / 
distant  r  centimeters  from  the  center  of  q.  Describe  a  sphere 
of  radius  r  with  its  center  at  the  center  of  q.  The  area  of  the 
surface  of  this  sphere  is  477T2,  and  the  electric  field  f  is  every- 
where at  right  angles  to  this  surface  and  everywhere  the  same  in 
value  at  the  surface.  Therefore  the  electric  flux  across  the  sur- 
face of  the  sphere  is  equal  to  ^irr2  x  /,  and  this  must  be  equal  to 
Bq  according  to  equation  (69),  where  B  has  the  value  given  in 
equation  (70).  Therefore  we  have 

Bq  =  477T2/ 
or 


I  So        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

B 


(70 


in  which  /  is  the  electric  field  intensity  in  volts  per  centimeter 
at  a  point  r  centimeters  from  a  concentrated  charge  of  q 
coulombs. 

Equation  (71)  applies  not  only  to  the  ideal  case  of  a  concen- 
trated charge,  but  also  to  the  case  in  which  the  charge  is  uniformly 
distributed  over  a  sphere. 

100.  Electrostatic  attraction  and  repulsion  of  concentrated 
charges.  —  Consider  a  concentrated  charge  q'  at  a  point  distant 
r  centimeters  from  another  concentrated  charge  q"  .  The  elec- 
tric field  intensity  at  q'  due  to  q"  is  given  by  equation  (71), 
namely, 


and  according  to  equation  (66),  the  force  with  which  this  field 
acts  upon  the  charge  q'  is  equal  to  the  product  q'fy  that  is, 

*<-£>*&  <?») 

47T      r1 

in  which  F  is  the  form  in  joule-units  *  with  which  the  two  con- 
centrated charges  q'  and  q"  repel  each  other  when  they  are  at 
a  distance  of  r  centimeters  apart,  q'  and  q"  being  expressed 
in  coulombs.  When  q'  and  q"  are  both  positive  or  both  nega- 
tive, the  force  .F  is  a  repulsion  ;  when  one  is  positive  and  the 
other  is  negative,  the  force  F  is  an  attraction. 

The  unit  of  charge  of  the  "electrostatic  system"  —  The  e.g.  s.  units  of  magnetic 
pole  strength,  magnetic  field  intensity,  electric  current  strength,  electromotive  force, 
electric  charge,  electrostatic  capacity,  etc.,  which  have  been  used  in  the  foregoing 
chapters  constitute  what  is  called  the  electromagnetic  system  of  units.  This  "  electro- 
magnetic system  "  starts  out  with  the  definition  of  a  unit  magnetic  pole  as  a  pole  of 
such  strength  that  it  will  exert  a  force  of  one  dyne  upon  an  equal  pole  at  a  distance 
of  one  centimeter.  The  so-called  electrostatic  system  of  units  starts  out  with  the  defi- 
nition of  the  unit  charge  as  a  charge  which  will  exert  a  force  of  one  dyne  upon  an  equal 

*  A  joule-unit  of  force  is  a  force  which  will  do  one  joule  of  work  in  pulling  a  body 
through  a  distance  of  one  centimeter  ;  it  is  equal  to  io7  dynes. 


ELECTRIC   CHARGE.     THE   CONDENSER.  l8l 

charge  at  a  distance  of  one  centimeter,  so  that,  if  the  two  charges   qf    and   qf/   are 
expressed  in  "electrostatic"  units,  and    F  in  dynes,  equation  (72)  becomes 

w 

It  is  convenient  to  call  the  electrostatic  system  of  units  the  Faraday  units  in  order 
to  distinguish  them  from  the  c.g.s.  units  which  are  employed  throughout  this  text. 
Thus,  the  Faraday  unit  of  electric  charge  is  a  charge  which  will  exert  a  force  of  one 
dyne  upon  an  equal  charge  at  a  distance  of  one  centimeter  in  air  ;  the  Faraday  unit 
of  electric  field  intensity  is  an  electric  field  of  such  strength  that  it  will  exert  a  force 
of  one  dyne  upon  a  body  which  carries  one  Faraday  unit  of  charge  ;  the  Faraday  unit 
of  electric  current  is  the  flow  of  one  Faraday  unit  of  charge  per  second  through  a  wire  ; 
the  Faraday  unit  of  magnetic  field  intensity  is  a  field  which  will  push  sidewise  with  a 
force  of  one  dyne  upon  each  centimeter  of  a  wire  carrying  one  Faraday  unit  of  electric 
current  ;  the  Faraday  unit  of  magnetic  pole  is  a  pole  of  such  strength  that  it  will  be 
acted  upon  by  a  force  of  one  dyne  in  a  magnetic  field  of  one  Faraday  unit  intensity  ; 
and  so  on.  The  electrostatic  system  of  units  (Faraday  units)  are  extensively  used  in 
advanced  treatises  on  Electricity  and  Magnetism.  In  this  text,  however,  the  electro- 
magnetic system  of  units  will  be  used,  that  is  to  say,  either  the  c.g.s.  units,  such  as 
the  abampere,  the  abohm,  the  abvolt,  the  abcoulomb,  the  abfarad,  etc.,  or  the  so- 
called  practical  units  such  as  the  ampere,  the  ohm,  the  volt,  the  coulomb,  the  farad, 
etc. 

Number  of  Faraday  units  of  charge  in  one  abcoulomb.  —  A  given  pair  of  charges 
attract  (or  repel)  each  other  with  a  definite  force  at  a  given  distance  apart  according 
to  equation  (72)>  'n  which  q  and  q'  are  expressed  in  coulombs  and  F  is  expressed 
in  joule-units  of  force.  If  F  is  expressed  in  dynes,  the  number  which  expresses  it 
must  be  ten  million  times  as  large,  so  that  the  right-hand  member  of  equation  (72) 
must  be  multiplied  by  io7  to  give  F  in  dynes.  The  force  with  which  two  concen- 
trated charges,  each  equal  to  one  abcoulomb  (io  coulombs),  would  repel  each  other 
at  a  distance  of  one  centimeter  apart  is  lOoB  /^TT  joule-units  of  force,  or  io9.#/47r 
dynes,  according  to  equation  (72)  ;  by  substituting  this  value  offeree  in  equation  (i), 
above,  placing  qf  =  q",  and  solving  for  q1  we  find  the  number  of  Faraday  units  of 
charge  in  one  abcoulomb,  namely,  3  X  Iol°-  This  result  is  equal  to  the  velocity  of 
light  in  air  in  centimeters  per  second.  See  Art.  146. 

101,  Electrostatic  attraction  of  parallel  plates.  —  Consider  two 
parallel  metal  plates  connected  to  a  battery  as  shown  in  Fig.  107. 
Let  a  be  the  area  of  each  plate,  x  their  distance  apart,  and  E 
the  electromotive  force  between  them.  The  two  plates  constitute 
a  condenser  of  which  the  capacity  is 

T     ka 


according  to  equation  (63),  where    k   is  the  inductivity  of  the 


182        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

dielectric  between  the  plates.  The  energy  of  the  charged  con- 
denser is 

•.W=l.?-.X  (ii) 

•  2    ka  ^  ' 

according  to  equation  (6$c). 

Let  the  battery  be  disconnected  and  the  plates  A  and  B  in- 
sulated so  that  q  cannot  change,  and  imagine  the  plates  to  be 
pulled  apart  so  as  to  increase  their  distance  by  the  amount  A;r 
(the  dielectric  being  a  fluid  like  air  or  oil).  Then  the  energy  of 
the  condenser  will  be  increased  by  the  amount 

A^=f.£.A,  (Hi) 

This  expression  for  the  increase  of  energy  of  the  charged  con- 
denser is  easily  derived  from  equation  (ii)  by  assuming  x  to  in- 
crease slightly.  This  increase  of  energy  of  the  charged  condenser 
comes  from  the  work  done  in  separating  the  plates  against  their 
mutual  attraction.  Let  F  be  the  force  of  attraction  of  the  plates, 
then  F-&X  is  the  work  done  against  their  mutual  attraction,  and 
this  is  equal  to  &W  in  equation  (iii)  so  that 

F.^*±.te 

2    ka 

or 


in  which  F  is  the  force  in  joule-units  with  which  two  parallel 
metal  plates  attract  each  other,  a  is  the  area  of  each  plate  in 
square  centimeters,  q  is  the  charge  on  each  plate  (positive  on 
one,  negative  on  the  other)  in  coulombs,  and  B  is  equal  to 
1.131  x  io13. 

It  is  noteworthy  that  the  force  of  attraction  is  independent  of 
the  distance  between  the  plates  and  inversely  proportional  to  the 
inductivity  of  the  dielectric,  the  charge  being  given.  The  plates 
are  supposed  to  be  very  large  in  comparison  with  the  distance 
between  them,  according  to  Art.  90. 


ELECTRIC   CHARGE.     THE   CONDENSER.  183 

Attraction  for  a  given  electromotive  force.  —  The  charge  q  in 
the  above  discussion  is  equal  to  the  capacity  of  the  condenser 
times  the  electromotive  force  between  the  plates,  according  to 

equation  (61),  that  is, 

I     kaE 


according  to  equations  (61)  and  (62).     Substituting  this  value  of 
q  in  equation  (73),  we  have 

I      ka& 

I    -     ;  £*a  I?-  (74) 

in  which  F  is  the  force  in  joule-units  with  which  two  metal  plates 
attract  each  other  in  air  or  oil,  E  is  the  electromotive  force  be- 
tween the  plates,  a  is  the  area  of  each  plate,  x  is  the  distance  be- 
tween the  plates  in  centimeters,  and  B  is  equal  to  1.131  x  io13. 
It  is  worthy  of  note  that  the  force  of  attraction  of  parallel  plates 
is  inversely  proportional  to  the  square  of  the  distance  between 
them  and  directly  proportional  to  the  inductivity  of  the  dielectric 
for  given  electromotive  force. 

102.  The  absolute  electrometer  is  an  arrangement  for  determining 
the  value  of  an  electromotive  force  by  measuring  the  force  of  at- 
traction of  parallel  metal  plates.  BALANCE 

The  value  of  the  electromotive 
force  is  calculated  from  equa- 
tion  (74)  when  k   (/£=  i,    for 
air),  a,  x,  and  B  are  known,       —  r~ 
and   F  is    observed   in   joule- 


units.     Figure    1 1 2  shows  the        f 
essential  features  of  the  abso- 
lute electrometer.     A  portion  of  area  a  of  the  upper  plate  is  hung 
from  one  end  of  a  balance  beam  so  that  the  force  with  which  this 
portion  is  attracted  by  the  lower  plate  may  be  counterpoised  by 
weights  placed  upon  the  scale  pan  and  thus  determined.     The 
stationary  portion  gg  of  the  upper  plate  completely  surrounds 
the  portion    a   and  is  called  the  guard  ring.     Equation  (74)  is 


1 84        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


true  only  for  plates  which  are  very  large  in  comparison  with  their 
distance  apart,  and  this  guard  ring  makes  it  possible  to  realize 
this  condition  approximately  without  making  the  moving  part  a 
of  the  upper  plate  inconveniently  large. 

The  value  of  the  constant  By  which  appears  in  equation  (62), 
Art.  90,  which  also  appears  in  the  equations  for  electrostatic 
attraction,  and  which  is  equal  to  the  amount  of  electric  flux  which 
emanates  from  one  coulomb  according  to  equation  (69),  may  be 
most  easily  determined  by  means  of  the  absolute  electrometer. 
A  known  electromotive  force  ,  E  is  connected  to  the  plates  of  an 


—  'fine  ^ 
Suspending 
wire 


sectional  view 


"y> 


top  view 

Fig.  1  13a.  Fte-  H3b. 

absolute  electrometer,  and  the  force  F  is  measured,  the  area  a 
of  the  movable  plate  and  the  distance  x  between  the  plates  be- 
ing known,  then  the  value  of  B  may  be  calculated  from  equa- 
tion (74),  k  being  equal  to  unity  for  air. 

103.  The  electrostatic  voltmeter  consists  essentially  of  a  fixed 
metal  plate  and  a  delicately  poised  or  suspended  metal  plate 
which  carries  a  pointer  which  plays  over  a  divided  scale.  The 


ELECTRIC   CHARGE.     THE   CONDENSER.  185 

electromotive  force  which  is  to  be  measured  is  connected  to  the 
two  plates  and  the  scale  is  numbered  so  as  to  indicate  the  value 
of  the  electromotive  force  directly. 

The  fixed  and  movable  plates  of  the  electrostatic  voltmeter  are 
usually  arranged  as  shown  in  Fig.  113^,  in  which  FF  and 
F'F  are  the  fixed  plates,  and  MM  is  the  movable  plate.  Figure 
113^  shows  an  electrostatic  voltmeter  designed  by  Lord  Kelvin 
for  measuring  electromotive  forces  ranging  from  80  to  140  volts. 
The  movable  plate  in  this  instrument  consists  of  a  large  number 
of  vanes  which  are  drawn  into  the  spaces  between  a  large  num- 
ber of  stationary  plates  essentially  as  in  Fig.  1  1  30. 

104.  Energy  and  tension  of  the  electric  field  in  air.  —  Consider 
the  charged  metal  plates  AA  and  BB  in  Fig.  107;  the  capac- 
ity of  the  plates  considered  as  a  condenser  is 

C---- 
~  B    x 

according  to  equation  (62),  where  B  =  1.  1  3  1  x  io13.  The  energy 
of  the  charged  condenser  A  ABB,  Fig.  107,  is 


according  to  equation  (65  £).  Therefore,  using  ijB^ajx  for  C, 
we  have 

w^—~  m 

"  2B        X 

This  energy  of  the  charged  condenser  resides  in  the  region  be- 
tween the  plates,  that  is,  in  the  electric  field.  The  volume  of  this 
region  is  ax.  Therefore,  dividing  both  members  of  equation  (i) 
by  ax,  we  find 

Energy  of  an  electric  field  in  )  I         -2  / 

joules  per  cubic  centimeter  J         ^B  \'j) 

in  which  /,  which  is  written  for  E[x,  is  the  intensity  of  the 
electric  field  between  the  plates  in  volts  per  centimeter. 

The  force  of  attraction  of  the  two  metal  plates    A  A    and    BB, 
Fig.  107,  is  given  by  equation  (74),  and  this  force  is  transmitted 


1 86        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

across  from  plate  to  plate  in  the  form  of  a  tension  of  the  electric 
field.  Therefore  dividing  both  members  of  equation  (74)  by  ay 
we  have  the  tension  of  the  electric  field  in  joule-units  offeree  per 
square  centimeter  of  section,  and  in  the  case  of  air  (k  equals 
unity),  we  have 

IB  "/2        W 


Tension  of  an  electric  field  in  joule-units  *)   ,. 
of  force  per  square  centimeter  of  section  j 


in  which  /,    which  is  written  for   £/x,    is  the  intensity  of  the 
electric  field  in  volts  per  centimeter. 

105.  Electric  potential.  —  The  two  heavy  black  circles  in  Fig. 
114  represent  two  long  parallel  metal  cylinders  one  of  which  is 


Fie.  114. 

positively  charged  and  the  other  of  which  is  negatively  charged, 
and  the  fine  curved  lines  (with  arrowheads)  represent  the  lines 
of  force  of  the  electric  field  in  the  region  between  the  charged 
cylinders.  The  intensity  of  the  electric  field  at  a  given  point  is 
so  many  volts  per  centimeter  parallel  to  the  lines  of  force  at  that 


ELECTRIC   CHARGE.     THE   CONDENSER.  187 

point.  Let  the  plane  of  the  paper  in  Fig.  114  be  a  horizontal 
plane,  and  imagine  a  hill  built  upon  this  plane  in  such  a  way 
that  its  slope  lines  as  seen  projected  upon  the  base  plane  coin- 
cide with  the  lines  of  force  in  Fig.  114.  If  the  height  of  this  hill 
is  measured  in  volts  then  its  slope  may  be  expressed  in  volts  per 
centimeter  at  each  point,  in  fact  its  slope  will  be  a  complete  rep- 
resentation of  the  electric  field  in  the  plane  of  Fig.  114.  The 
height,  at  a  point,  of  an  imagined  hill  w/tose  slope  is  everywhere 
equal  to  the  electric  field  is  called  the  electric  potential  at  that 
point.  The  heavy  curved  lines  ppp  in  Fig.  1 14  are  the  contour 
lines,  or  lines  of  equal  level,  on  the  potential  hill  which  is  im- 
agined to  be  built  as  described  above.  The  potential  is  there- 
fore the  same  at  every  point  along  each  of  the  heavy  curved  lines 
and  these  lines  are  therefore  called  lines  of  equipotential. 

The  above  example  refers  to  the  distribution  of  electric  field  in 
two  dimensions,  and  in  this  case  the  potential  hill  may  be  actually 
constructed  as  a  geometrical  hill.  In  general,  however,  this  is 
not  possible,  that  is  to  say,  it  is  not  possible  to  construct  a  geo- 
metrical representation  of  the  potential  hill.  A  clear  idea  of  po- 
tential in  this  general  case  may  be  obtained  as  follows  :  Imagine 
any  given  distribution  of  electric  field,  the  electric  field  surround- 
ing a  charged  sphere,  for  example,  and  imagine  the  region  sur- 
rounding the  sphere  to  vary  in  temperature  from  point  to  point  in 
such  a  way  that  the  temperature  gradient  (degrees  per  centimeter) 
at  each  point  may  be  equal  to  the  electric  field  (volts  per  centime- 
ter) at  that  point.  Then  the  temperature  at  each  point  represents 
what  is  called  the  electric  potential  at  that  point.  In  this  ex- 
ample of  the  field  surrounding  a  charged  sphere,  the  lines  of  force 
are  radial  straight  lines  and  any  surface  drawn  so  as  to  be  at 
each  point  at  right  angles  to  the  lines  of  force  is  a  surface  of  equi- 
potential. 

In  order  to  completely  establish  the  value  of  the  electric  poten- 
tial at  different  points  in  space,  a  region  of  zero  potential  must  be 
arbitrarily  chosen.  Then  the  potential  at  any  other  point  is 
equal  to  the  electromotive  force  E  between  the  arbitrarily  chosen 


1 88        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

region  of  zero  potential  and  the  given  point  The  product  Eg 
is  equal  to  the  work  required  to  carry  charge  q  from  the  region 
of  zero  potential  to  the  given  point,  and  it  is  therefore  equal  to 
the  potential  energy  of  the  charge  q  when  it  is  placed  at  the 
given  point. 

The  idea  of  potential  is  important  in  the  mathematical  theory 
of  electricity  and  magnetism,  but  jts  use  by  students  who  are 
beginning  the  study  of  the  subject  of  electricity  and  magnetism 
tends  to  turn  the  attention  away  from  physical  realities. 

PROBLEMS. 

115.  During  0.03  second  a  charge   of    15   coulombs  passes 
through  a  circuit.     What  is  the  average  value  of  the  current  dur- 
ing this  time  ?     Ans.   500  amperes. 

116.  Suppose  the  strength  of  a  current  in  a  circuit  to  increase 
at  a  uniform  rate  from  zero  to  50  amperes  in  3  seconds.     Find 
the  number  of  coulombs  of  charge  carried  through  the  circuit  by 
the  current  during  the  3  seconds.     Ans.   75  coulombs. 

117.  A  condenser  of  which  the  capacity  is  known  to  be  5 
microfarads,  is  charged  by  a   Clark  standard  cell  of  which  the 
electromotive  force  is  1.434  volts  and  then  discharged  through  a 
ballistic  galvanometer.     The  throw  of  the  ballistic  galvanometer  is 
observed  to  be  15.3  scale  divisions.     What  is  the  reduction  factor 
of  the  galvanometer  ?     Ans.  469  x  i  o~9  coulombs  per  division. 

118.  A  condenser  of  unknown  capacity  is  charged  by  10  Clark 
cells  in  series,  giving  an  electromotive  force  of  14.34  volts,  and 
then  discharged  through  the  ballistic  galvanometer  specified  in 
problem  117;  the  throw  of  the  ballistic  galvanometer  is  observed 
to  be   1 8. 6  divisions.     What  is  the  capacity  of  the  condenser? 
Ans.  0.608  microfarad. 

119.  An  electromotive  force  acting  on  a  condenser  increases  at 
a  uniform  rate  from  zero  to  100  volts  during  an  interval  of  1/200 
of  a  second.     The  capacity  of  the  condenser  is  20  microfarads. 
Find  the  value  of  the  current  during  the    1/200  of  a  second. 
Ans.  0.2  ampere. 


ELECTRIC   CHARGE.     THE   CONDENSER.  189 

120.  An  alternating  electromotive  force  which  is  represented 
by  the  ordinates  of  the  zigzag  line  in  Fig.  1 1 5  acts  on  a  circuit 
which  contains  a  condenser.  The  resistance  of  the  circuit  is  neg- 
ligible, and  the  capacity  of  the  condenser  is  20  microfarads. 


Axis  of 


*~ygooim3T" 

Fig.    115. 

Plot  the  curve  of  which  the  ordinates  represent  the  successive 
instantaneous  values  of  the  current. 

121.  Two  parallel  metal  plates  at  a  fixed  distance  apart  with 
air  between  are  charged  as  a  condenser  and  discharged  through 
a  ballistic  galvanometer.     The  plates  are  then  submerged  in  tur- 
pentine and  again  charged  and  discharged  through  a  ballistic  gal- 
vanometer.     The  charging  electromotive  force  is  the  same  in 
each  case  and  the  throw  of  the  ballistic  galvanometer  is  observed 
to  be  7.6  divisions  in  the  first  instance  and  16.7  in  the  second 
instance.      Find  the  inductivity  of  the  turpentine.     Ans.   2.2. 

122.  The  metal  core  and  metal  sheath  of  a  submarine  cable 
are  separated  by  an  insulating  layer  of  gutta-percha  and  they 
constitute  the  two  plates  of  a  condenser.     One  mile  of  a  subma- 
rine cable  has  a  capacity  of  0.06  microfarad.     What  is  the  ca- 
pacity of  100  miles  of  the  cable?     Ans.  6  microfarads. 

123.  A  condenser  is  to  be  built  up  of  sheets  of  tin  foil  12  cen- 
timeters X  I  5  centimeters.     The  overlapping  portions  of  the  sheets 
are   12  centimeters  x   12  centimeters.     The  sheets  are  separated 
by  leaves  of  mica  0.05  centimeter  thick.     How  many  mica  leaves 
and  how  many  tin  foil   sheets  are  required  for  a  one-microfarad 
condenser  ?     Assume  the  inductivity  of  the  mica  to  be  equal  to 
6.     Ans.   Mica,  655  ;  tin  foil,  656. 

124.  A  condenser  is  made  of  two  flat  metal  plates  separated  by 
air.     Its  capacity  is  0.003  microfarad.     Another  condenser  has 


190        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

plates  twice  as  wide  and  twice  as  long.  These  plates  are  sepa- 
rated by  a  plate  of  glass  (inductivity  5)  which  is  four  times  as 
thick  as  the  air  space  in  the  first  condenser.  What  is  the  capacity 
of  the  second  condenser?  Ans.  0.015  microfarad. 

125.  Two  metal  plates,  100  centimeters  x  100  centimeters,  are 
separated  by  2,  centimeters  of  air.     This  condenser  is  charged  by 
a  battery  having  an  electromotive  force  of  2,000  volts.     What  is 
its  energy  in  joules  ?     Ans.  0.000884  joule. 

126.  A  flat  glass  plate,  inductivity  5,  size  100  centimeters  x  IOO 
centimeters  x  2   centimeters,   is  slid  between   the    metal    plates 
specified  in  problem  125,  the  battery  being  left  connected  to  the 
metal  plates.     What  is  the  energy  of  the  condenser  after  the 
glass  is  in  place  ?     Ans.   0.00442  joule. 

127.  The  2,ooo-volt  battery  is  disconnected  from  the  metal 
plates  specified  in  problem  1 26  after  the  glass  is  in  place  and  the 
metal  plates  are  thoroughly  insulated.     The  glass  plate  is  then 
withdrawn,  the  whole  charge  being  left   on   the  metal  plates. 
What  is  the  electromotive  force  between  the  metal  plates  after 
the  glass  plate  is  withdrawn  ?     What  is  the  energy  of  the  con- 
denser after  the  glass  plate  is  withdrawn  ?     How  much  has  the 
energy  been  increased  by  withdrawing  the  glass  ?     How  much 
force  was   necessary  to   withdraw  the    glass,   ignoring  friction, 
weight,  etc.  ?     (Assume  that  the  glass  is  withdrawn  sidewise,  not 
cornerwise.)     Ans.    10,000  volts,  0.0221   joule,  0.01768  joule, 
1,768  dynes. 

128.  The  air  condenser  specified  in  problem   125  is  charged 
with  2,000  volts,  the  battery  is  disconnected  and  the  plates  are 
then  moved  to   a  distance  3   centimeters    apart,  charge  on  the 
plates  remaining  unchanged.     What  is  the  electromotive  force 
between  the  plates  after  the  movement  ?     What  is  the  increase 
of  energy  due  to  the  movement?     How  much  force  was  neces- 
sary to  produce  the  movement,  ignoring  friction,  weight,  etc.? 
Ans.   3,000  volts,  0.000442  joule,  4,420  dynes. 

129.  What  is  the  intensity  of  the  electric  field  between  two 


ELECTRIC   CHARGE.     THE   CONDENSER. 


parallel  metal  plates,  1 5  centimeters  apart,  the  electromotive  force 
between  the  plates  being  2 5,000  volts?  Ans.  1,667  volts  Per 
centimeter. 

130.  A  large  metal  ball  is  placed  in  the  uniform  electric  field 
between  the  plates  specified  in  problem  129  and  the  ball  is  acted 
upon  by  a  force  of  2.5  dynes.     What  is  the  charge  on  the  ball  in 
coulombs  ?     Ans.    1 5  x  IO~U  coulomb. 

131.  A  spark  gauge  s  is  connected  with  two  metal  plates  AB 
as  shown  in  Fig.  1 1 6.     A  sheet  of  paraffined  paper  0.002  of  an 

inch  in  thickness  is  placed 

^  m  *  •  ^i 

between  A  and  B  and  the 
spark  gap  s  is  slowly  in- 
creased until  the  paraffined 
paper  is  punctured.  The 
spark  gap  is  then  meas- 
ured and  found  to  be 
equal  to  o.  1 6  centimeter. 
The  radius  of  the  spheres 
of  the  spark  gauge  is  0.25 
centimeter.  Find  the  value 
of  the  electromotive  force 
required  to  puncture  the 
piece  of  paper.  Ans. 
6,950  volts. 

133.  What  is  the  inten- 
sity of  the  electric  field  in 
volts  per  centimeter  at  a 
distance  of  200  centimeters 

from  the  center  of  a  sphere  upon  which  0.0002  coulomb  of  charge 
is  uniformly  distributed  ?  Ans.  4,500  volts  per  centimeter. 

134.  Find  the  maximum  charge  that  can  be  held  on  a  sphere 
200  centimeters  in  diameter  in  air  on  the  assumption  that  the 
electric  field  intensity  at  the  surface  of  the  sphere  cannot  exceed 
22,000  volts  per  centimeter  without  breaking  the  air  down  elec- 
trically.    Ans.  0.000979  coulomb. 


to  electric  machine 

Fig.  116. 


192        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

135.  A  very  long  metal  cylinder,  10  centimeters  in  diameter, 
is  charged  and  the  amount  of  charge  on  each  centimeter  of  length 
of  the  cylinder  is  3  x.Jp~12  coulomb.      Find  the  intensity  of  the 
electric  field  at  a  point  distant  50  centimeters  from  the  axis  of  the 
cylinder.     Ans.  0.108  volt  per  centimeter. 

136.  Find  the  maximum  charge  that  can  be  held  on  each  unit 
length  of  the  cylinder  specified  in  problem  1 3  5  on  the  assumption 
that  the  electric  field  intensity  at  the  surface  of  the  cylinder  can- 
not exceed  22,000  volts  per  centimeter  without  breaking  down 
the  air.     Ans.  62  x  io~9  coulomb. 

137.  A  liquid  covers  a  horizontal  plane  at  a  uniform  depth  of 
10  centimeters.     At  a  point  in  this  plane  there  is  a  hole  through 
which  the  liquid  flows  at  the  rate  of  15   cubic  centimeters  per 
second.     Find  the  direction  and  magnitude  of  the  velocity  of  the 
liquid  on  the  plane  at  a  point  distant   r   centimeters  from  the 
center  of  the  hole.     Ans.  o.2$8/r  centimeters  per  second. 

138.  Two  parallel  metal  plates  each  one  centimeter  in  diameter 
are  placed  in  pure  distilled  water  at  a  distance  of  10  centimeters 
apart  and  an  electromotive  force  of  100  volts  is  connected  to  the 
two  plates.     Find  the  force  in  dynes  with  which  the  plates  attract 
each  other,  the  inductivity  of  the  water  being  equal  to  90.     Ans. 
32x1 0~*  dynes. 

139.  The  entire  region  throughout  a  room  is  a  uniform  electric 
field  directed  vertically  upwards  and  its  intensity  is  2,000  volts 
per  centimeter,     (a)    Choosing  the  floor  as  the  region  of  zero 
potential,  what  is  the  potential  at  a  point   1 50  centimeters  above 
the  floor  ?     (£)  What  kind  of  lines  are  the  lines  of  force,  straight 
or  curved,  and  in  what  direction  ?     (c)  What  kind  of  surfaces  are 
the  surfaces  of  equipotential,  plane  or  curved,  and  in  what  direc- 
tion do  these  surfaces  lie  ?     Ans.  (a)  300,000  volts.     (&)  Straight 
lines,  perpendicular  to  floor,     (c)  Plane  surfaces,  parallel  to  floor. 

140.  Given  two  parallel  metal  plates  1 5  centimeters  apart  to 
which  a   io,ooo-volt  battery  is  connected.     Imagine  a  line    X 
drawn    straight   from   plate   to   plate.      Choose   the    negatively 


ELECTRIC   CHARGE.     THE   CONDENSER.  193 

charged  plate  as  the  region  of  zero  potential  and  plot  a  curve  of 
which  the  abscissas  are  distances  measured  along  the  line  X  and 
of  which  the  ordinates  represent  the  values  of  the  potential  at 
points  along  this  line. 

141.  An  inclined   plane  is  viewed  from  above.     A  series  of 
contour  lines  and  of  slope  lines  are  drawn  upon  the  plane.     Make 
a  diagram  showing  the  appearance  of  these  lines  as  projected 
upon  the  base  plane. 

142.  A  circular  cone  is  viewed  from  above  and  a  series  of  con- 
tour lines  and  slope  lines  are  seen  projected  upon  the  base  plane 
of  the  cone.     Draw  a  diagram  showing  the  appearance  of  these 
lines  on  the  base  plane. 


CHAPfER   VIII. 
THE   PHENOMENA   OF   ELECTROSTATICS. 

106.  The  voltaic  cell  (or  dynamo)  versus  multiplying  devices  for 
the  production  of  large  electromotive  forces.  —  A  locomotive  engi- 
neer, knowing  that  an  ordinary  locomotive  can  exert  about  1 5,000 
pounds  of  draw-bar  pull,  and,  wishing  to  observe  the  behavior  of 
a  bar  of  steel  when  subjected  to  a  stretching  force  of  1 50,000 
pounds,  might  arrange  to  use  ten  locomotives  hitched  together  to 
exert  the  desired  force ;  but  if  the  bar  of  steel  should  break,  then 
the  dormant  energy  of  the  locomotives  would  come  into  action, 
about  1 0,000  actual  horse-power  would  have  to  be  taken  care  of, 
and  a  terrible  wreck  would  probably  be  the  result.  The  value  of 
a  locomotive  lies  in  the  fact  that  it  can  continue  to  pull  even  when 
the  thing  it  pulls  on  yields,  as  it  were,  at  a  speed  of  60  miles  per 
hour ;  but  for  exerting  a  large  force  upon  a  thing  which  does  not 
yield  rapidly,  some  sort  of  a  force-multiplying  device,  such  as  a 
screw  or  a  lever,  is  more  convenient  and  incomparably  cheaper 
and  safer  than  a  battery  of  locomotives. 

An  electrician,  knowing  that  an  ordinary  dry  cell  has  an  elec- 
tromotive force  of  about  1.5  volts,  and,  wishing  to  observe  the 
effects  when  the  air  between  two  metal  plates  is  subjected  to  a 
high  electromotive  force,  might  think  of  connecting  100,000  dry 
cells  in  series  to  exert  1 50,000  volts ;  but  if  the  air  should  break 
down,  then  the  dormant  energy  of  the  battery  would  come  into 
action,  about  1,000  actual  horse-power  would  have  to  be  taken 
care  of,  the  apparatus  would  in  all  likelihood  be  destroyed,  and 
if  the  body  of  the  electrician  should  by  accident  become  a  portion 
of  the  battery  circuit,  he  would  be  instantly  killed.  The  value  of 
the  battery  or  dynamo  lies  in  the  fact  that  it  can  continue  to  push 
even  when  the  circuit  upon  which  it  pushes  yields  at  a  "speed"  of 
many  amperes  ;  but  for  exerting  a  large  electromotive  force  across 

194 


THE   PHENOMENA   OF   ELECTROSTATICS.  195 

a  fairly  good  insulator  which  does  not  yield  *  to  any  great  extent, 
some  sort  of  an  electromotive-force-multiplying  device  is  more 
convenient  and  incomparably  cheaper  and  safer  than  a  battery  of 
dynamos  or  voltaic  cells. 

107.  The  alternating-current  transformer  or  induction  coil.  — 

One  method  for  multiplying  electromotive  force  is  by  means  of 
the  alternating  -  current  transformer, 
which  is  strictly  analogous  to  a  me- 
chanical lever  pinned  to  a  very  massive 
movable  body,  instead  of  to  a  rigid 
fulcrum,  as  shown  in  Fig.  1 1 7.  Such 
a  fulcrum  gives  way  under  a  steady 
force,  but  it  is  sufficiently  immovable 
under  the  action  of  an  alternating  force, 
that  is,  a  force  which  is  repeatedly  re- 
versed in  direction.  The  alternating- 
current  transformer  or  induction  coil 
cannot  be  used  to  produce  a  large  steady  electromotive  force. f 

The  mechanical  analogue  of  the  alternating-current  transformer. 
—  A  lever  Aa,  Fig.  1 1 7,  of  negligible  mass,  is  attached  to  a  very 
massive  body  M  by  a  pin  connection.  An  alternating  force  acts 
on  the  end  A  causing  it  to  oscillate  back  and  forth  along  the 
dotted  line  with  great  alternating  velocity  /,  and  the  end  a  of 

*  An  electromotive  force  of  100,000  volts  connected  to  two  metal  plates  each  one 
meter  square  with  a  plate  of  flint  glass  between  them  two  centimeters  thick  would 
produce  about  one  ten-million-millionth  of  an  ampere  through  the  plate  of  glass. 
See  table  of  specific  resistances  in  Art.  14. 

f  Accessory  devices  may,  however,  be  used  in  conjunction  with  an  alternating-cur- 
rent transformer  to  produce  an  approximately  steady  uni-directional  electromotive  force. 
One  of  these  devices,  the  mercury-arc  rectifier,  is  strictly  analogous  to  the  mechanical 
device  called  a  ratchet  which  permits  a  body  to  move  in  one  direction  only  ;  and 
another  device  is  the  commutator  which  is  analogous  to  the  crank  which  converts  the 
alternating  force  of  a  locomotive  engine  into  a  pulsating,  uni-directional,  draw-bar 
pull,  which  may  become  fairly  steady  in  value  if  the  moving  locomotive  is  very  mas- 
sive. It  is  hot  an  exaggeration  to  say  that  no  one  can  understand  the  mercury-arc 
rectifier  or  the  commutator  or  any  other  electrical  or  magnetic  device  or  phenomenon 
unless  he  can  reduce  it  in  his  mind  to  its  mechanical  equivalent.  See  the  Preface  to 
this  volume. 


196        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 


the  lever  oscillates  back  and  forth  with  a  small  alternating 
velocity  i.  If  the  motion  of  the  end  a  of  the  lever  is  opposed 
by  a  considerable  frictional  resistance,  requiring  a  large  alter- 
nating force  E  to  overcome  it,  then  a  certain  small  alternating 
force  e  must  act  on  the  end  A  of  the  lever  to  produce  the 
required  alternating  force  E.  4 

One  who  is  familiar  with  the  action  of  the  alternating  current  transformer  may  fol- 
low out  this  mechanical  analogue  in  all  of  its  details.  The  alternating  velocity  of  the 
end  A  corresponds  to  the  primary  current  /',  and  the  alternating  velocity  of  the  end 
a  corresponds  to  the  secondary  current  ///.  Immovability  of  the  end  a  corresponds 
to  open  secondary  circuit,  and  entire  freedom  of  motion  of  the  end  a  corresponds  to 
short-circuited  secondary.  If  the  mass  M  were  infinite,  the  end  A  could  not  move 
at  all  when  the  end  a  is  fixed  (open  secondary),  but  if  the  mass  M  is  finite,  then  a 
given  alternating  force  Ef  acting  on  the  end  A  would  cause  some  motion  of  the  end 
At  even  if  the  end  a  were  rigidly  fixed,  and  this  motion  of  the  end  A  corresponds 
to  the  magnetizing  current  of  the  transformer. 

108.  The  electrical  doubler.  —  The  device  which  is  most  fre- 
quently used  for  building  up  intense  electrical  fields  (high  elec- 
tromotive forces)  is  shown  in  its  simplest  form  in  Figs.  1 1 8  to  121. 


B 


Fig.  118. 


Fig.  1 1 9. 


It  is  desired  to  build  up  a  very  intense  electrical  field  between  a 
metal  plate  A  A  and  one  side  of  a  hollow  metal  vessel  BB.  A 
metal  ball  C,  called  a  carrier,  is  attached  to  an  insulating  handle 


THE   PHENOMENA   OF   ELECTROSTATICS. 


197 


by  means  of  which  it  can  be  brought  into  contact  with  the  point 
p  and  then  pushed  into  the  interior  of  BB  and  brought  into 
contact  with  BB,  repeatedly.  Each  time  the  carrier  touches  the 
point  /  it  receives  a  certain  amount  of  charge  from  the  battery 
b,  or  in  other  words,  a  bundle  of  lines  of  force  comes  into  exist- 
ence between  C  and  A  A  as  shown  in  Fig.  1 1 8.  As  the  car- 


A 


Fig.  120. 


Fig.  121. 


rier  is  moved  into  the  hollow  vessel  BB,  the  bundle  of  lines  of 
force  trends  as  shown  in  Fig  119,  and  work  has  to  be  done  to 
move  the  carrier  against  the  pull  of  these  lines  of  force.  As  the 
carrier  is  moved  into  BB  the  lines  of  force  from  C  to  AA 
are  cut  in  two,  as  it  were,  one  after  another,  by  the  metal  wall  at 
w9  the  portions  of  the  lines  of  force  which  pass  from  C  to  AA, 
as  shown  in  Fig.  1 20,  are  then  obliterated  by  bringing  C  into 
contact  with  B,  and  the  carrier  C  is  left  entirely  neutral  as 
shown  in  Fig.  121.  Each  repetition  of  the  above  movements  of 
the  carrier  C  "  strings  "  an  additional  bundle  of  lines  of  force 
from  A  to  B  and  thus  increases  the  intensity  of  the  electrical 
field  between  A  and  B. 

The  production  of  a  very  large  electromotive  force  between 
A    and    B   in  Figs.  1 1 8  to  121  by  the  to  and  fro  motion  of  the 


198        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 


Fig.  122. 


carrier  C  is  somewhat  analogous  to  the  production  of  an  intense 
stress  in  a  steel  block  AABB,  Fig.  122,  by  stretching  small 

rubber  bands,  like  R,  and  placing 
them  over  the  block,  one  after  an 
other.  The  simplest  mechanical  an- 
alogue of  the  electric  doubler,  how- 
ever, is  the  building  up  of  intense 
stresses  in  a  drum  by  winding  upon 
it  a  string  or  wire  under  consider- 
able tension.* 

In  the  electrical  doubler  which  is 
represented  in  Figs  1 1 8  to  121,  the 
carrier  C  receives  charge  repeatedly 

from  a  battery  b.  The  frictional  electrical  machine  and  the  influ- 
ence electrical  machine  employ  the  principle  of  the  electrical 
doubler.  In  these  machines,  however,  the  carrier  is  charged  not  by 
a  battery  of  voltaic  cells,  but  by  either  of  two  peculiar  electrical 
processes,  namely,  (a)  charging  by  contact  and  separation,  or  (b) 
charging  by  influence.^  These  two  processes  are  described  in 

*A  very  interesting  example  of  this  action  is  described  on  page  339  of  the  third 
volume  of  Lord  Kelvin's  Popular  Lectures  and  Addresses.  When  Lord  Kelvin  was 
carrying  out  his  first  experiments  on  deep  sea  sounding,  the  long  piano-steel  wire 
which  was  used  was  wound 
upon  a  heavy  metal  drum,  and 
the  stress  in  the  drum  became 
great  enough  to  bend  it  out  of 
shape. 

|  The  simplest  device  com- 
bining charging  by  influence 
and  the  doubling  action  which 
is  described  in  connection  with 
Figs.  118  to  121  is  shown  in 
Fig.  123.  The  hollow  metal 
vessels  A  and  B  have  a  cer- 
tain amount  of  charge  to  begin 
with.  Two  flat  metal  carriers 

C  and  D  each  having  an  insu-  P. 

lating   handle,  are  placed   be- 
tween A  and  B  and  brought  into  contact  with  each  other  as  shown.     The  result  is 
that  the  lines  of  force  from   A  to  B   arrange  themselves  as  shown  in  the  figure,  and 
then  the  carriers  C  and  D  may  be  separated  from  each  other,  carrier  C  being  moved 


THE   PHENOMENA   OF   ELECTROSTATICS.  199 

the  following  articles,  and  in  order  to  obtain  a  clear  understanding 
of  the  frictional  electric  machine,  of  the  Toepler-Holtz  electrical 
machine,  and  of  the  Wimshurst  electrical  machine,  it  is  important 
to  keep  in  mind  the  fact  that  every  one  of  these  machines  involves 
the  principle  of  the  electrical  doubler,  inasmuch  as  the  carrier  or 
carriers  pass  between  two  conducting  bodies  to  both  of  which  they 
give  up  their  charges,  so  that  these  two  conducting  bodies  take 
the  place  of  the  hollow  vessel  BB  in  Figs.  118  to  121. 

109.  Charging  by  contact  and  separation.  —  The  production  of 
electric  charge  by  the  rubbing  together  of  certain  substances  is 
one  of  the  most  familiar  of  the  phenomena  of  electricity.  When 
a  cat  is  stroked  with  the  hand  in  a  dry  room,  the  cat's  fur  and 
the  hand  become  oppositely  charged,  and  the  crackling  sound 
which  is  produced  is  due  to  the  production  of  minute  electrical 
sparks  which  may  be  seen  if  the  room  is  dark.  A  hard  rubber 
comb  becomes  strongly  charged  when  it  is  passed  through  very 
dry  hair,  and  the  comb  will  attract  small  bits  of  paper  or  pith. 
When  pencil  marks  are  erased  from  a  very  dry  piece  of  paper  by 
means  of  a  rubber  eraser,  the  paper  becomes  charged  and  it 
clings  to  the  drawing  board  or  table. 

Two  substances  when  brought  into  contact  always  tend  to 
settle  to  a  state  of  equilibrium  in  which  electric  lines  of  force 
pass  from  one  substance  to  the  other  across  the  very  thin  air 
space  between  them.  Thus,  two  flat  plates  of  zinc  and  copper 
settle  to  equilibrium  with  an  electromotive  force  of  about  0.9 
volt  between  them,  so  that  the  intensity  of  the  electric  field  in  the 
very  narrow  space  between  the  plates  may  be  several  thousands 
volts  per  centimeter.  If  the  plates  are  thoroughly  insulated  and 
moved  apart,  the  electric  field  intensity  (volts  per  centimeter)  re- 

into  the  interior  of  B  and  brought  into  contact  with  B,  and  carrier  D  being  moved  into 
the  interior  of  A  and  brought  into  contact  with  A.  The  result  is  that  the  bundle 
of  lines  of  force  from  A  to  C  in  Fig.  123  is  stretched  across  from  A  to  B,  and 
the  bundle  of  lines  of  force  from  D  to  B  is  stretched  across  from  A  to  B  thus 
increasing  the  total  number  of  lines  of  force  from  A  to  B.  The  revolving  doubler 
of  Lord  Kelvin  is  a  mechanical  device  for  performing  the  operations  here  described, 
the  carriers  C  and  D  being  mounted  upon  a  rotating,  insulated  arm. 


200        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 


mains  unaltered,  so  that  the  electromotive  force  between  the 
plates  may  be  increased  to  several  thousands  of  volts.  Thus,  the 
very  fine  lines  in  Figs.  124  and  125  represent  the  electric  field 


copper 


Fig.  124. 


zinc 


copper 


zinc 


Fig.  125. 


between  the  copper  and  zinc  plates  when  they  are  close  together 
and  after  they  have  been  separated  to  a  considerable  distance. 

In  order  to  produce  an  intense  electric  field  by  separating  two 
metal  plates,  the  plates  must  be  very  flat,  and  they  must  be 
separated  in  such  a  way  as  to  avoid  a  lingering  contact  between 
them.  When  both  of  the  substances  are  good  insulators,  how- 
ever, they  always  retain  their  charges  (one  positive  and  the  other 
negative)  when  they  are  moved  apart,  and  the  intervening  region 
becomes  an  intense  electric  field.  This  phenomenon  is  called 
charging  by  contact  and  separation.  In  order  to  bring  sealing 
wax  and  fur,  or  glass  and  silk  into  intimate  contact,  vigorous 
rubbing  is  necessary,  and  therefore  charging  by  contact  and 
separation  is  frequently  spoken  of  as  charging  by  friction. 

To  understand  the  phenomenon  of  charging  by  contact  and 
separation  it  is  important  to  keep  in  mind  that  the  charging  is 
done  by  contact  (no  one  knows  exactly  how),  and  that  the  crea- 
tion of  an  intense  electrical  field  throughout  a  large  region  is  ac- 
complished by  separation.  In  this  case  the  electrical  field  is 
wound  up,*  as  it  were,  by  pulling  the  charged  surfaces  apart,  and 
the  work  done  in  pulling  the  charged  surfaces  apart  against  their 
force  of  attraction  (tension  of  the  lines  of  force)  is  the  work  that 
goes  to  establish  the  field  in  the  larger  and  larger  region  between 
the  receding  surfaces. 

*  In  the  sense  of  winding  up  a  spring  so  as  to  put  it  under  stress. 


THE    PHENOMENA   OF    ELECTROSTATICS. 


201 


110.  The  f fictional  electric  machine.  —  This  machine  consists  of 
a  rotating  glass  disk  DD,  Fig.  126,  the  various  parts  of  which 
come  in  succession  into  intimate  contact  with  two  leather 
cushions  AA  which  are  impregnated  with  an  amalgam  of  tin, 
zinc  and  mercury.  The  surface  of  the  glass  plate  as  it  leaves 
these  cushions  is  left  highly  charged  with  positive  electricity,  and 


side  view 


top  view 

Fig.  126. 

the  cushions  are  left  negatively  charged.  The  negative  charge 
flows  into  the  insulated  conductor  N  which  is  connected  to  the 
cushions  by  means  of  the  metal  springs  55,  and  the  positive 
charge  is  carried  on  the  surface  of  the  glass  disk  to  the  collecting 
combs  CC  whence  it  flows  into  the  insulated  conductor  P. 
Two  silk  aprons  ppt  one  on  each  side  of  the  rotating  disk,  tend 


202        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


Fig.  127. 


to  prevent  the  escape  of  the  positive  charge  from  the  surface  of 
the  disk.* 

111.  Charging  by  influence.  -3  The  simplest  example  of  charg- 
ing by  influence  is  that  which  is  described  in  connection  with  Fig. 

123.  Charging  by  influence  is 
essentially  the  cutting  of  lines 
of  force  in  two  by  two  sheets 
of  metal  so  that  the  ending  of 
the  lines  of  force  on  one  sheet 
constitute  a  new  negative 
charge  and  the  beginning  of 
the  lines  of  force  on  the  other 
sheet  constitute  a  new  positive 
charge.  Let  A,  Fig.  127,  be  a 
charged  body  from  which  lines 
of  force  emanate.  When  a 
metal  ball  B  is  brought  near  to 
A,  the  lines  of  force  converge 
upon  one  side  of  B  and  diverge  from  the  other  side  as  shown  in 
the  figure ;  if  a  second  metal  ball  C  is  brought  into  contact  with 
B,  as  shown  in  Fig.  128,  then  the  lines  of  force  converge  upon 
B  and  diverge  from  C,  and  the  two  balls  B  and  C  retain  their 
charges  when  they  are  separated  and  removed  to  a  distance  from 
A.  This  operation  is  called  charging  by  influence t  and  it  results 
in  the  production  of  equal  amounts  of  positive  and  negative  elec- 
tricity (on  B  and  C  respectively)  while  the  original  influencing 
charge  on  A  is  undiminished.  Charging  by  influence  is  exem- 
plified by  the  operation  of  the  electrophorus. 

112.  The  electrophorus  is  a  device  for  the  production  of  charge 
by  influence.     It  consists  of  a  rosin  or  hard  rubber  plate   D, 

*  The  frictional  electric  machine  involves  the  principle  of  the  electric  doubler,  but 
it  is  not  worth  while  to  examine  minutely  into  the  manner  in  which  the  lines  of  force 
are  drawn  out  as  it  were  and  "strung"  across  from  P  to  N,  as  the  various  parts 
of  the  glass  plate  leave  the  cushions  A  A.  The  above  account,  which  is  based  on  the 
idea  that  positive  and  negative  electricities  are  two  fluid-like  substances,  is  sufficiently 
intelligible  for  present  purposes. 


THE   PHENOMENA   OF   ELECTROSTATICS.  203 

Fig.    129,  which  has  been  electrified  (negatively)  by  rubbing  it 
with  a  piece  of  fur  or  flannel,  and  a  metal  disk   M   with  an 


Fig.  128. 

insulating  handle  H.  When  the  metal  disk  is  brought  near  to 
the  negatively  charged  plate  of  rosin  and  touched  with  the  finger 
it  is  left  with  a  charge  of  positive  electric- 
ity, and  this  charge  remains  on  M,  when 
M  is  removed  to  a  distance  from  D.  This 
operation  may  be  repeated  indefinitely.* 

113.  Influence  electrical  machines. — The  .. ^^ 

D  ^  '-'^-ZZ:?Z%%\ 

electrophorus  is  the  simplest  arrangement 

for  the  generation  of  charge  by  influence. 

If  the  metal  carrier    M    of  the  electrophorus  is  thrust  into  a 

hollow  metal  vessel  and  touched  to  its  walls,  it  gives  its  entire 

charge  to  the  hollow  vessel,  whatever  the  previous  charge  on  the 

vessel  may  be,  and  thus  it  is  possibe  to  generate  any  desired 

*  The  description  here  given  of  the  operation  of  the  electrophorus  is  really  inade- 
quate. The  metal  pan  which  contains  the  rosin  plate  plays  an  important  part  in  the 
operation  of  the  electrophorus  as  is  evident  from  the  fact  that  the  electrophorus  does 
not  operate  satisfactorily  when  the  metal  pan  is  insulated  from  the  floor  and  walls  of 
the  room  by  being  placed  upon  an  insulating  support. 


204       ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

electromotive  force  between  the  hollow  metal  vessel  and  the 
walls  of  the  room.  In  the  Toepler-Holtz  machine  and  in  the 
Wimshurst  machine,,  metal  carriers  are  fixed  to  a  rotating  glass 
disk  or  disks  so  that  at  one 'part  of  their  path  these  carriers 
become  charged  by  influence  and  at  another  part  of  their  path 
they  pass  between  two  pieces  of  metal  which  act  like  the  hollow 
metal  vessel  in  Figs.  118  to  121,  thus  combining  the  principle 
of  the  electrical  doubler  with  the  principle  of  the  electrophorus. 
The  inducing  charge  (which  corresponds  to  the  charge  on  the 
rosin  plate  of  the  electrophorus)  in  the  Toepler-Holtz  machine 
and  in  the  Wimshurst  machine  is  generated  by  the  machine  itself. 
Reversibility  of  influence  machines. — The  Toepler-Holtz  ma- 
chine and  the  Wimshurst  machine  may  be  used  as  electric  gen- 
erators as  described  below,  in  which  case  they  must  be  supplied 
with  mechanical  power  and  they  deliver  electrical  charge  at  high 
electromotive  force;  or  they  may  be  used  as  electric  motors  in 
which  case  they  must  be  supplied  with  electric  charge  at  high 
electromotive  force  from  some  outside  source,  and  they  deliver 
mechanical  power.  Thus,  a  very  large  Toepler-Holtz  machine 
driven  at  high  speed  may  deliver  a  steady  current  of  o.ooi  of  an 
ampere  (one  thousandth  of  a  coulomb  of  charge  per  second)  at  an 
electromotive  force  of,  say,  100,000  volts.  This  corresponds  to 
an  output  of  100  watts  of  power,  and  if  the  friction  losses  in  a 
second  similar  machine  are  very  small,  the  second  machine  may 
be  driven  as  a  motor. 

114.  The  Toepler-Holtz  machine. — A  general  view  of  the 
Toepler-Holtz  machine  is  shown  in  Fig.  130.  It  is  difficult  to 
show  in  a  diagram  the  essential  features  of  such  a  machine  in 
which  the  carriers  are  arranged  on  a  glass  disk.  Figure  131 
shows  a  possible  form  of  Toepler-Holtz  machine  in  which  the 
carriers  are  fixed  to  a  rotating  glass  cylinder  which  is  surrounded 
by  a  stationary  glass  cylinder  upon  which  the  "  inductors " 
AA  and  BB  (which  carry  the  inducing  charges)  are  supported. 
The  neutralizing  rod  is  a  stationary  metal  rod  with  metal  brushes 
at  its  ends,  and  the  figure  shows  the  metal  brushes  2  and  4  in 


THE    PHENOMENA   OF   ELECTROSTATICS. 


20; 


contact  with  the  metal  buttons  which  project  from  two  of  the 
carriers.     The  result  is  that  these  two  carriers  become  charged 


under  the  influence  of  the  positive  and  negative  charges  on   AA 
and   BB,    the  upper  carrier  being  negatively  charged  and  the 


Fig.  131. 


lower  carrier  being  positively  charged.     The  rotation  of  the  inner 
cylinder  then  moves  the  carriers  in  the  direction  of  the  curved 


206        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

arrows  until  the  carriers  come  under  brushes  5  and  6  where  they 
part  with  a  portion  of  their  charges,  thus  replenishing  inducing 
charges  on  AA  and  BB.  The  carriers  are  then  moved  into 
the  space  between  A  A  and  A'  A'  on  the  one  hand  and  into  the 
space  between  BB  and  Bf Bf  on  the  other  hand  where  they 
come  into  contact  with  the  brushes  I  and  3,  thus  giving  up  the 
remainder  of  their  charges  to  the  metal  terminals  TT  of  the 
machine.  When  a  spark  is  formed  between  the  metal  terminals 
TT,  the  bodies  A' A'  and  B'Bf  become  completely  discharged, 
but  the  induced  charges  on  AA  and  BB  remain  and  the 
machine  continues  to  operate. 

The  Toepler-Holtz  machine  is  self-exciting,  that  is  to  say,  the 
extremely  minute  electromotive  forces  due  to  the  contact  of  the 
metal  brushes  with  the  metal  buttons  on  the  carriers  are  sufficient 
to  start  the  operation  of  charging  by  influence,  and  the  action  of 


Fig.  132. 

the  machine  is  then  rapidly  intensified  by  the  doubling  action 
which  takes  place. 

115.  The  Wimshurst  machine.  —  A  general  view  of  the  Wims- 
hurst  machine  is  shown  in  Fig.  132.  It  consists  of  two  oppo- 
sitely rotating  glass  disks  on  each  of  which  a  number  of  metal 


THE   PHENOMENA   OF   ELECTROSTATICS.  2O/ 

carriers  are  fixed.  Stationary  neutralizing  rods  are  placed  one 
on  each  side  of  the  machine,  each  inclined  at  an  angle  of  approxi- 
mately 45°  to  the  horizontal,  and  the  charge  on  one  disk  as  it 
travels  towards  the  collectors  serves  as  the  inducing  charge  for 
the  other  disk. 

The  inducing  action  of  the  Wimshurst  machine  may  be  ex- 
plained as  follows :  Figure  133  represents  two  glass  plates   AB 


Fig.  133. 

and  CD.  One  of  these  plates  is  charged  as  indicated  by  the 
plus  signs,  and  the  lines  of  force  from  this  charge  converge  upon 
the  metal  point  P  which  is  at  one  end  of  a  neutralizing  rod. 


+     -r        +         +        +       +        -h          -r 
Fig.  134. 

The  electric  field  in  the  neighborhood  of  P  is  sufficiently  intense 
to  break  down  the  air  between  /'and  the  glass  plate  CD,  thus 
leaving  negative  charge  on  the  glass  plate  CD  as  shown  in 

Fig.  134. 

The  small  portion  of  the  surface  of  CD  which  faces  the  point 
P  is  thus  negatively  charged  and  the  amount  of  charge  on  this 
small  portion  is  numerically  equal  to  the  amount  of  positive 
charge  on  the  larger  part  of  AB  from  which  emanate  the  lines 
of  force  that  have  been  broken  down  between  P  and  CD.  If 
the  plate  CD  is  moved  to  the  left  in  Fig.  134,  fresh  lines  of 
force  crowd  into  the  space  between  the  point  P  and  the  plate 
CD,  they  continue  to  break  down  as  in  the  first  instance,  and  the 
entire  surface  of  CD,  as  it  moves  out  from  under  the  point 


208        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Pt  is  left  much  more  strongly  charged  than  the  plate  AB. 
The  plate  AB  may  itself  be  charged  by  moving  it  in  front  of  a 
point  Pf  under  the.  inducing  action  of  CD  as  shown  in  Fig. 


f    f  t 


Fig.  135. 


135.  Under  these  conditions,  the  charges  on  AB  and  CD  will 
grow  more  and  more  intense  until  checked  by  the  rapidly  increas- 
ing leakage  from  the  surfaces  of  the  plates.  The  negative  charge 
on  CDy  Fig.  135,  after  it  has  passed  beyond  the  point  P'9  and 


Fig.  136. 


the  positive  charge  on    AB,    after  it  has  passed  beyond  the  point 
P,    may  be  collected  and  used  for  any  purpose. 

Figure  136  shows  the  essential  features  of  a  complete  Wims- 


THE   PHENOMENA   OF    ELECTROSTATICS. 


209 


Fig.  137. 


hurst  machine  consisting  of  two  coaxial  glass  cylinders  rotating 
in  opposite  directions.  The  negative  charges  on  both  cylinders 
are  collected  by  the  double  metal  comb  on  the  left  as  the  rotating 
cylinders  pass  between  the  prongs  of  the  comb,  and  the  positive 
charges  of  both  cylinders  are  collected  by 
the  double  comb  on  the  right. 

116.  Electroscopes.  —  An  electroscope  is 
a  device  for  indicating  the  existence  of  an 
electric  charge,  or  for  detecting  an  electric 
field. 

The  pith  ball  electroscope  consists  of  a 
gilded  ball  of  pith  suspended  by  a  silk  thread. 
The  presence  of  an  electric  field  in  a  given 
region  may  be  shown  by  charging  the  pith  ball,  and  noting  the 
force  which  acts  upon  it  when  it  is  placed  in  the  given  region, 
the  direction  of  the  field  being  indicated  by  the  direction  of  the 
force  which  acts  upon  the  ball. 

A  pith  ball  may  be  hung  alongside  of  a  body  of  metal,  as 
shown  in  Fig.  137.  If  the  body  of  metal  is  charged,  a  portion 
of  the  charge  is  given  to  the  ball,  and  the  lines  of  force  which 
emanate  from  the  ball  pull  it  outwards  from  the  body  as  shown 
in  the  figure. 

The  essential  features  of  the  gold  leaf  electroscope  are  shown  in 
Fig.  138.  A  metal  rod  R  is  supported  in  the  top  of  a  glass  case 
cc  by  means  of  an  insulating  plug,  a  metal  disk  D  is  fixed  to 
the  upper  end  of  the  rod,  and  two  gold  leaves  are  hung  side  by 
side  from  the  lower  end  of  the  rod.  The  glass  case  cc  serves  to 
protect  the  gold  leaves  from  air  currents.  The  sides  of  cc  are 
lined  with  strips  of  metal  foil  ff,  and  these  pieces  of  metal  should 
be  connected  to  earth.  When  the  disk,  rod  and  leaves  are  charged, 
the  leaves  are  pulled  apart  by  the  lines  of  force  which  emanate 
from  the  leaves  and  terminate  on  the  strips  ff  as  shown  in  Fig. 
139.  This  figure  shows  the  instrument  without  the  enclosing 
case  for  the  sake  of  clearness. 


210        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 


Fig.  138. 


Fig.  1 39. 


Tlie  behavior  of  a  gold  leaf  electroscope  when  a  charged  body  is 
brought  near  to  the  plate  D  is  as  follows  :  (i)  When  the  elec- 
troscope has  no  initial  charge,  some  of  the  lines  of  force  pass 
from  the  charged  body  into  the  disk  and  then  spread  out  from 
the  leaves  to  the  strips  fft  causing  the  leaves  to  diverge.  If 
the  charged  body  is  removed  the  electroscope  becomes  neutral 
and  the  leaves  fall  together.  If,  while  the  charged  body  is  near 
D,  the  disk  or  rod  is  touched  with  the  finger,  the  lines  of  force 
passing  out  from  the  leaves  cease  to  exist,  and  the  leaves  fall  to- 
gether. If  now,  the  charged  body  is  removed,  the  lines  of  force 
passing  into  the  disk  from  the  charged  body  spread  over  the  disk, 
rod  and  leaves,  and  the  electroscope  is  left  charged,  as  indicated 
by  the  divergence  of  the  leaves.  This  operation,  called  charging 
by  influence,  is  explained  more  fully  in  Art.  1 1 1. 

(2)  When  the  electroscope  has  an  initial  charge,  say  a  positive 
charge,  then  a  positively  charged  body  brought  near  to  D  pushes 
the  initial  charge  down  into  the  leaves,  as  it  were,  and  the  diver- 
gence of  the  leaves  is  increased.  If  a  negatively  charged  body 
is  brought  near  to  D,  the  positive  charge  on  the  leaves  is  pulled 
up  into  the  disk,  as  it  were,  by  the  attraction  of  the  negative 
charge  on  the  body,  and  the  divergence  of  the  leaves  is  decreased. 
If  the  negatively  charged  body  is  brought  nearer,  the  leaves  will 
come  together  ;  and  if  the  negatively  charged  body  is  brought 
still  nearer  the  leaves  will  again  diverge. 


THE   PHENOMENA   OF   ELECTROSTATICS.  211 

The  behavior  of  a  positively  charged  electroscope  when  a 
negatively  charged  body  is  brought  near  to  it,  is  the  same  as  its 
behavior  when  it  is  negatively  charged  and  a  positively  charged 
body  is  brought  near  to  it 

117.  Electric  charge  resides  wholly  on  the  surface  of  a  charged 
conductor.     Electrical  screening.  —  The  electrostatic  phenomena 
exhibited  by  charged  conductors  are  precisely  the  same  whether 
the  bodies  be  solid  or  hollow ; 
and,  if  the  bodies  be  hollow, 
no  effect  of  the  charges  can  be 
detected  inside  of  them   how- 
ever thin  their  walls  may  be. 
The  lines  of  force  of  the  electric 
field  end  at  the  surface  of  the 
charged  conductor  or,  in  other 
words,  the  electric  charge  re- 
sides wholly  on  the  surface  of  a 
charged  conductor. 

A  conducting  shell,  such  as 
a  metal  box,  screens  its  interior 

completely,  so  that  no  action  of  any  kind  reaches  the  interior  from 
charged  bodies  outside.*  Thus,  a  hollow  metal  ball  C,  Fig. 
140,  screens  its  interior  completely.  The  lines  of  force  which 
touch  the  shell  C  end  at  its  surface.  The  ending  on  C  of  the 
lines  of  force  from  A  is  negative  charge  and  the  beginning  on 
C  of  the  lines  of  force  which  reach  B  is  positive  charge. 

The  fact  that  electrical  field  cannot  penetrate  into  a  substance 
like  a  metal  shows  that  such  substances  cannot  sustain  the  pecul- 
iar kind  of  stress  which  constitutes  electrical  field  any  more  than 
a  fluid  can  sustain  the  kind  of  stress  that  exists  in  a  stretched 
steel  wire. 

Mechanical  analogue  of  electrical  screening.  —  Consider  a  solid 
body  B,  Fig.  141,  entirely  separated  from  the  surrounding  solid 
by  an  empty  space  eee.  Stress  and  distortion  of  the  surrounding 

*  This  is  not  strictly  true  when  the  outside  conditions  are  changing  rapidly. 


212        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


solid  cannot  affect  B  in  any  way,  and  conversely  stress  and  dis- 
tortion of  B  cannot  affect  the  surrounding  solid  because  the 
-  ,  empjy  space  is  incapable  of  transmit- 
ting stress.  This  empty  space  in  its 
behavior  towards  mechanical  stress  is 
analogous  to  a  conducting  substance 
in  its  behavior  towards  electrical 
stresses  (electrical  field). 

118.  A  charged  conductor  shares  its 
charge  with  another  conductor  which 
is  brought  into  contact  with  it. —  Fig- 
ure 142  shows  the  lines  of  force  in  the  neighborhood  of  a  charged 
conductor  A.  When  another  conductor  B  is  brought  into  con- 
tact with  Ay  the  lines  of  force  arrange  themselves  as  shown  in 
Fig.  143.  The  charge  which  was  originally  on  A  spreads  over 
A  and  B,  as  indicated  by  the  ending  of  the  lines  of  force. 


Fig.  141, 


Fig.  142. 


Fig.  143. 


119.  Faraday's  experiment.  —  A  charged  body  B,  Fig.  144^ 
is  lowered  into  a  metal  vessel  and  the  opening  of  the  vessel  is 
closed  with  a  metal  lid.  As  the  body  is  lowered  into  the  vessel, 
each  line  of  force  that  emanates  from  B  is  cut  in  two,  as  it  were, 
by  the  wall  of  the  vessel  so  that,  when  B  is  entirely  enclosed 
by  the  vessel,  as  many  lines  of  force  emanate  from  the  external 
surface  of  the  vessel  as  from  the  body  B,  and  all  the  lines  of 


THE    PHENOMENA   OF   ELECTROSTATICS. 


213 


force  which  emanate  from  B  terminate  on  the  inner  surface  of 
the  vessel.  Therefore,  if  +q  is  the  amount  of  charge  on  B, 
—  q  is  the  amount  of  charge  on  the  inner  surface  of  the  vessel 


silk 
thread 


Fig.  144. 


Fig   145. 


Fig.  146. 


Fig.  147. 


and    -\-  g   is  the  amount  of  charge  on  the  external  surface  of  the 
vessel  in  Fig.  145. 

After  the  body    B   has  been  completely  enclosed  by  the  metal 
vessel  as  shown  in  Figs.  145,  146,  and   147,  the  distribution  of 


214        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  electrical  field  outside  of  the  vessel  does  not  depend  in  any 
way  upon  the  position  of  the  body  B  inside  the  vessel,  and,  if 
the  body  B  is  brought  into  contact  with  the  wall  of  the  vessel, 
the  lines  of  force  which  emanate  from  B  disappear,  no  charge  is 
left  on  B  and  no  charge  is  left  on  the  inner  surface  of  the 
vessel.  $ 

120.  Giving  up  of   entire  charge   by  one  body  to  another. — 

When  the  body  B,  Figs.  144,  145,  146,  and  147,  is  lowered 
into  the  vessel  and  allowed  to  touch  the  walls  of  the  vessel  it  loses 
all  of  its  charge  and  remains  without  charge  when  removed  from 
the  vessel,  and  the  charge  on  the  outside  of  the  vessel  is  equal  to 
and  of  the  same  sign  as  the  original  charge  on  B.  The  body  B 
may  thus  be  said  to  give  up  its  entire  charge  to  the  vessel. 

121.  Convective  discharge  and  disruptive  discharge.  —  Consider 
the  positive  and  negative  charges  at  the  ends  of  a  bundle  of  lines 
of  force.      In  order  that  these  charges  may  disappear,  it  is  neces- 
sary that  the  lines  of  force  be  annihilated.     This  may  be  accom- 
plished by  the  moving  of  the  charged  surfaces  towards  each  other 
until  they  are  coincident,  or  the  dielectric  which  sustains  the  elec- 
trical stress  may  break  down.     In  the  former  case,  we  have  what 
is  known  as  convective  discharge,  and,  in  the  latter  case,  we  have 
what  is  known  as  disruptive  discharge. 

Convective  discharge  is  to  some  extent  analogous  to  the  re- 
lieving of  a  stretched  rubber  band  by  allowing  its  ends  to  move 
towards  each  other.  Disruptive  discharge  is  somewhat  analo- 
gous to  the  relief  of  a  stretched  rubber  band  by  rupture. 

Examples. — (a)  Two  metal  plates  A  A  and  BB  in  Fig.  106, 
being  disconnected  from  the  battery,  might  be  discharged 
(the  electric  field  be  made  to  disappear)  by  moving  the  plates 
together. 

(b)  The  transfer  of  charge  by  a  moving  ball,  as  described  in 
Art.  94,  is  convective  discharge.  The  ball  gathers  in  the  ends 
of  a  bundle  of  lines  of  force  when  it  touches  one  plate  and  it 
shortens  these  lines  until  they  disappear  as  it  moves  across  to  the 


THE    PHENOMENA   OF   ELECTROSTATICS. 


215 


other  plate.  Figures  148  to  151  show  the  successive  aspects  of 
the  electric  field  while  the  ball  is  moving  once  across  from  plate 
to  plate. 

(c)  The  electromotive  force  between  the  two  metal  balls    A 
and    B,    Fig.  no,  may  be  increased  until  the  intervening  dielec- 


Fig.  148- 


Fig.  150. 


Fig.  149. 


Fig.  151. 


trie  breaks  down,  causing  the  formation  of  an  electric  spark.  An 
•electric  spark  is  a  conducting  path,  like  a  wire,  and  its  effect  is  to 
completely  discharge  the  two  balls  A  and  B. 

122.  Progress  of  the  electric  spark.  —  Let  A  and  B,  Fig. 
152,  be  two  metal  balls  upon  which  electric  charge  has  accumu- 
lated until  the  intensity  of  the  electric  field  has  reached  the 


2l6        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

breaking  point  of  the  intervening  dielectric.  The  rupture  of  the 
dielectric  starts  in  the  region  of  greatest  electric  stress,*  as  indi- 
cated by  the  short  thick  line  projecting  from  the  surface  of  A 
in  the  figure.  Along  the  line  of  this  rupture  the  dielectric  is  a 


Fig.  152. 

good  conductor,  and  the  lines  of  force  on  all  sides  move  sidewise 
into  the  rupture  as  indicated  by  the  arrows,  producing  a  greatly 
intensified  electric  field  at  the  end  of  the  rupture  so  that  the  rup- 
ture extends  further  and  further  until  it  reaches  B. 

This  extension  of  an  electric  rupture  or  spark  through  a  region 
in  which  the  intensity  of  the  electric  field  is  originally  much  be- 
low the  breaking  value  of  the  dielectric  is  analogous  to  the  fol- 
lowing :  A  pane  of  glass  is  slightly  bent  and  then  scratched  near 
one  edge  so  as  to  start  a  crack.  The  effect  of  this  crack  is  to 
greatly  intensify  the  stress  in  the  glass  at  the  end  of  the  crack 
and  the  crack  therefore  quickly  runs  across  the  pane. 

When  the  electric  rupture  has  extended  itself  across  from  A 
to  B  in  Fig.  152,  a  conducting  line  or  path  is  established  from 
A  to  B,  and  all  of  the  charge  on  A  and  B  disappears,  that 
is  to  say,  the  electric  field  between  A  and  B  disappears. 

123.  The  brush  discharge.  —  The  discharge  in  air  from  a  body 
of  metal  which  stands  at  a  distance  from  surrounding  bodies  is  in 
some  respects  different  in  character  from  the  spark  discharge  be- 
tween two  oppositely  charged  conductors  which  are  not  too  far 
apart.  Figure  153  represents  the  lines  offeree  spreading  out 

*  This  rupture  always  starts  in  air  at  the  surface  of  the  positively  charged  ball,  un- 
less the  surface  of  the  other  ball  is  much  more  sharply  curved. 


THE   PHENOMENA   OF   ELECTROSTATICS. 


217 


from  a  positively  charged  metal  ball.  If  the  ball  is  sufficiently 
charged  the  electric  field  near  its  surface  reaches  the  breaking 
point  of  the  dielectric,  and  the  rupture  starts  as  described  in  Art. 


Fig.  153. 


122,  but  in  this  case  the  rupture  very  soon  extends  into  a  region 
where  the  field  was  originally  very  much  less  intense  than  at  the 
surface  of  the  ball,  and  such  lines  of  force  as  have  moved  side- 


Fig.  154. 


wise  into  the  rupture  and  have  partially  (that  is,  through  a  portion 
of  their  length)  broken  down,  now  form  in  a  widely  divergent 
bundle  from  the  end  of  the  rupture  as  shown  in  Fig.  1 54  (com- 


218        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


pare  Fig.  154  with  Fig.  152).  The  result  is  that  the  rupture 
divides  into  many  branches  which  penetrate  into  the  surrounding 
air  in  the  form  of  a  tree  or  brush.  This  type  of  discharge  is  called 
the  brush  discharge,  and  it  is '"most  readily  formed  in  a  region 
where  the  lines  of  electric  force  are  widely  divergent,  as  near  a 
pointed  projection  on  a  charged  conductor.  The  brush  discharge 
forms  more  readily  on  a  positively  charged  conductor  than  on  a 
negatively  charged  conductor,  and  the  positive  brush  is  very  dif- 
ferent in  character  from  the  negative  brush. 

124.  Electric  discharge  from  metallic  points.  — A  body  of  metal 
which  has  a  sharp  point  can  scarcely  be  charged  at  all,  because 
of  the  fact  that  a  very  slight  charge  on  the  body  produces  a  very 
intense  electric  field  in  the  neighborhood  of  the  sharp  point,  the 
lines  of  force  in  this  region  break  down,  and  the  lines  of  force  be- 
come detached  from  the  conductor,  ending  upon  charged  portions 
of  the  surrounding  air.  Thus,  Fig.  155^  represents  a  metal  ball 


Fig.  155. 


with  a  sharp  metal  point,  and  Fig.  155^  represents  the  state  of 
affairs  after  the  air  has  broken  down  in  the  neighborhood  of  the 
sharp  point  where  the  electric  field  is  very  intense. 

The  tension  of  the  lines  of  force  cd  in  Fig.  155$  pulls  the 
positively  charged  air  away  from  the  point,  forming  a  blast  of 
air.  If  the  ball  is  connected  to  an  electric  machine  so  as  to  be 
continually  supplied  with  charge,  new  lines  of  force  continually 
replace  those  that  are  broken  down  and  a  continuous  blast  of  air 
is  produced  which  is  sometimes  strong  enough  to  blow  out  a 
candle. 


THE   PHENOMENA   OF   ELECTROSTATICS. 


2I9 


Fig.  156. 


Figure  156  shows  the  bent  end  of  a  metal  rod  with  a  sharp 
point  at  P.  When  the  lines  of  force  emanate  from  all  parts  of 
the  rod  as  shown  in  the  figure,  the  total  force  acting  on  the  rod 
is  zero,  if  it  is  at  some  distance  from  surrounding  objects.  When, 
however,  the  lines  of  force  near 
the  point  break  down,  they  no 
longer  pull  on  the  rod,  therefore 
the  pull  due  to  the  lines  of  force 
at  b  is  unbalanced,  and  the  rod 
is  acted  upon  by  a  force  pulling 
it  to  the  left.  The  electric  whirl 
is  an  arrangement  of  pointed  rods  bent  as  shown  in  Fig.  157, 
and  mounted  on  a  pivot  on  an  insulating  stand.  When  this 
arrangement  is  connected  to  an  electric  machine,  it  is  set  into 
very  rapid  rotation  by  the  unbalanced  pull  of  the  lines  of  force 

which  emanate  from  the  portions 
b  of  the  rods,  as  shown  in  Fig. 
156. 

125.  The  mechanical  theory  of 
electricity  and  the  atomic  theory 
of  electricity. — The  study  of  elec- 
tricity and  magnetism  as  repre- 
sented in  the  foregoing  chapters 
(with  the  exception  of  several  mat- 
ters which  are  discussed  in  Chap- 
ter I)  is  independent  of  any  con- 
sideration of  the  nature  of  the  physical  action  which  leads  to 
the  production  of  electromotive  force  by  a  voltaic  cell  or  dynamo ; 
it  is  independent  of  any  consideration  of  the  nature  of  the  physical 
action  which  constitutes  an  electric  current  in  a  wire ;  it  is  inde- 
pendent of  any  consideration  of  the  nature  of  the  disturbance  which 
constitutes  a  magnetic  field ;  and  it  is  independent  of  any  con- 
sideration of  the  nature  of  the  disturbance  or  stress  which  consti- 
tutes an  electric  field.  This  kind  of  study  of  electricity  and  mag- 
netism may  very  properly  be  called  electro-mechanics. 


Fig.  157. 


220.        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  science  of  mechanics  is,  in  a  broad  sense,  the  study  of  those 
phenomena  which  depend  upon  the  mutual  actions  of  bodies  in 
bulk.  Thus  the  study. of  the  behavior  of  a  railway  car  under  the 
combined  action  of  the  pull  of  the  locomotive  and  the  drag  of  the 
track  belongs  to  the  science  of  mechanics.  The  study  of  the  beha- 
vior of  a  magnet  in  the  neighborhood  of  an  eleqtric  circuit  belongs 
to  the  science  of  mechanics.  The  study  of  the  behavior  of  two 
electrically-charged  bodies  belongs  to  the  science  of  mechanics. 

Simple  mechanics  is  the  study  of  ordinary  bodies  at  rest  or  in 
motion,  and  one  of  the  most  important  ideas  in  the  science  of 
simple  mechanics  is  the  idea  offeree  ;  but  the  science  of  mechanics 
is  not  concerned  with,  and  it  sheds  no  light  upon  the  question  as 
to  the  exact  physical  nature  of  force.  Thus,  the  science  of 
mechanics  is  not  concerned  with  the  question  as  to  the  nature  of 
the  action  which  takes  place  in  a  gas  causing  the  gas  to  exert  a 
force  on  a  piston  ;  the  science  of  mechanics  is  not  concerned  with 
the  question  as  to  the  nature  of  the  action  which  takes  place  in 
the  material  of  a  stretched  spring  causing  the  spring  to  exert  a 
force ;  the  science  of  mechanics  is  not  concerned  with  the  nature 
of  the  action  between  the  earth  and  a  heavy  weight  causing  the 
earth  to  exert  a  force  on  the  weight ;  the  science  of  mechanics  is 
not  concerned  with  the  nature  of  the  action  which  takes  place 
between  two  bodies  which  slide  over  each  other  and  which  leads 
to  the  production  of  the  force  of  friction.  It  is  sufficient  for  the 
science  of  mechanics  that ,  these  actions  are  what  may  be  called 
states  of  permanency  of  the  respective  systems.  Thus,  to  say  that 
a  gas  in  a  given  cylinder  pushes  with  a  force  of  100  pounds  on  a 
piston,  is  to  specify  a  definite  result  of  a  definite  condition  or  state 
of  the  gas,  and  it  is  this  definite  result  that  is  important  rather 
than  the  details  of  the  physical  action  which  is  taking  place  in  the 
gas.  In  fact,  the  science  of  mechanics  owes  its  existence  to  the  legiti- 
macy and  usefulness  of  the  idea  of  force  irrespective  of  the  nature 
of  the  physical  processes  upon  which  force  action  depends* 

*A  very  remarkable  discussion  "On  the  Scope  of  Mechanical  Explanation  and 
on   the   Idea   of  Force"   is  given   in    Appendix  B,   pages    268-288,    of  Larraor's 
and  Matter,  Cambridge,  1900. 


.      THE   PHENOMENA   OF    ELECTROSTATICS.  221 

t 

Similarly,  it  is  sufficient  for  the  science  of  electro-mechanics 
that  the  physical  actions  which  underlie  electromotive  force,  elec- 
tric current,  magnetic  field  and  electric  field  are  what  may  be 
called  states  of  permanency ;  thus,  to  say  that  a  current  of  ten 
amperes  flows  through  a  wire  is  to  specify  a  definite  effect  of  a 
definite  condition  or  state  of  the  wire,  and  it  is  the  correlation 
between  the  definite  condition  and  the  definite  effect  that  is  im- 
portant rather  than  the  details  of  the  physical  action  which  is 
taking  place  in  the  wire.  In  fact,  the  science  of  electro-mechanics 
owes  its  existence  to  the  legitimacy  and  usefulness  of  the  ideas  of 
electromotive  force,  electric  current,  magnetic  field  and  electric  field, 
irrespective  of  the  nature  of  the  physical  actions  upon  which  these 
various  things  depend. 

The  superficial  character*  of  the  science  of  simple  mechanics  and 
of  the  science  of  electro-mechanics  may  be  further  exemplified  as 
follows  :  Let  us  consider,  on  the  one  hand,  the  idea  of  tensile 
strength.  A  piece  of  steel  is  broken  by  a  tension  of  120,000 
pounds  per  square  inch,  but  the  exact  character  of  the  action 
which  takes  place  in  the  steel  and  which  constitutes  the  tension 
of  the  steel,  and  the  exact  character  of  the  physical  action  which 
takes  place  in  the  engine  or  motor  which  operates  the  testing 
machine  and  subjects  the  rod  of  steel  to  tension  are  not  matters 
for  consideration.  Indeed,  nothing  at  all  is  known  fundamentally 
as  to  the  physical  action  which  constitutes  the  tension  of  a  bar  of 
steel.  Let  us  consider,  on  the  other  hand,  the  idea  of  dielectric 
strength.  A  plate  of  glass  is  broken  down  by  an  electric  field  of 
95,000  volts  per  centimeter,  but  the  exact  nature  of  the  stress 
which  constitutes  the  electric  field  and  the  exact  character  of  the 

*  What  has  been  said  above  concerning  the  scope  of  mechanics  may  be  exemplified 
as  follows  :  Simple  mechanics  is  concerned  with  the  correlation  of  measurable  effects, 
such  as  the  relationship  between  the  size  of  a  beam  and  the  load  it  can  carry,  the  size 
of  a  fly-wheel  and  the  work  it  can  do  when  it  is  stopped,  the  thickness  and  diameter 
of  a  boiler  shell  and  the  pressure  which  it  can  withstand,  the  size  of  a  submerged  body 
and  the  buoyant  force  which  acts  upon  it,  the  size  and  shape  of  an  air  column  and  its 
number  of  vibrations  per  second,  and  so  on.  It  is  evident  that  such  relations  as  these 
do  not  involve  any  consideration  of  the  intimate  nature  of  the  physical  actions  which 
are  taking  place. 


222        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

physical  action  which  enables  a  voltaic  cell  or  dynamo  to  exert 
the  required  electromotive  force  are  not  matters  for  consideration, 
although,  as  a  matter  .of  fact,  much  more  is  known  concerning 
the  nature  of  electric  field  than  is  known  concerning  the  nature 
of  mechanical  stresses  in  substances  like  steel. 

The  science  of  mechanics,  as  stated  above,  deals  with  those 
phenomena  which  depend  upon  the  mutual  actions  of  bodies  in 
bulk.  The  phenomena  of  chemical  action  and  those  physical 
phenomena  which  have  to  do  with  the  minute  details  of  physical 
processes,  however,  have  been  studied  heretofore  almost  solely 
on  the  basis  of  the  atomic  theory.  Thus,  nearly  the  whole  of 
chemistry  is  based  on  the  atomic  theory  ;  the  kinetic  theory  of 
gases  is  a  branch  of  the  atomic  theory ;  the  theory  of  crystal  for- 
mation is  a  branch  of  the  atomic  theory ;  the  study  of  the  phe- 
nomena of  electrolysis  is  a  branch  of  the  atomic  theory  ;  and  the 
study  of  the  phenomena  of  the  discharge  of  electricity  through 
gases  is  a  branch  of  the  atomic  theory. 

126.  Electrons  and  ions. — The  loss  of  electricity  from  a 
charged  body  has  long  been  known  to  be  due  in  part  to  a  leak- 
age of  the  electricity  through  the  surrounding  air  and  in  part  to 
a  leakage  of  the  electricity  through  the  insulating  supports  of  the 
charged  body.  That  is  to  say,  the  air  conducts  electricity  to 
some  extent.  The  electrical  conductivity  of  the  air  is  ordinarily 
extremely  small,  but  there  are  a  number  of  influences  which 
cause  the  air  (or  any  gas)  to  become  a  fairly  good  electrical  con- 
ductor. Thus,  a  gas  becomes  a  fairly  good  conductor  when  its 
temperature  is  raised  above  a  certain  point ;  gas  which  is  drawn 
from  the  neighborhood  of  a  flame  or  electric  arc,  or  from  the 
neighborhood  of  glowing  metal  or  carbon,  is  a  fairly  good  con- 
ductor ;  gas  which  has  been  drawn  from  a  region  through  which 
an  electric  discharge  has  recently  passed  is  a  fairly  good  conduc- 
tor ;  and  the  passage,  through  a  gas,  of  ultra-violet  light,  of 
Roentgen  rays,  or  of  the  radiations  from  radio-active  substances, 
causes  the  gas  to  become  a  fairly  good  conductor.  The  conduc- 


THE    PHENOMENA   OF    ELECTROSTATICS.  223 

tivity  which  is  imparted  to  a  gas  by  these  various  agencies  may 
be  destroyed  by  filtering  the  gas  through  glass-wool  or  by  plac- 
ing the  gas  for  a  few  moments  between  electrically  charged  metal 
plates.  This  effect  of  filtration  seems  to  show  that  the  conduc- 
tivity of  a  gas  is  due  to  something  which  is  mixed  with  the  gas, 
and  the  effect  of  the  electric  field  (between  two  charged  plates) 
shows  that  this  something  is  charged  with  electricity  and  moves 
under  the  action  of  the  field.  "  We  are  thus  led  to  the  conclu- 
sion that  the  conductivity  of  a  gas  is  due  to  electrified  particles 
mixed  up  with  the  gas,  some  positive,  some  negative.  We  shall 
call  these  electrified  particles  ions  and  the  process  by  which  a  gas 
is  made  into  a  conductor  we  shall  call  the  process  of  ionization"* 

The  electron^  is  a  negatively  charged  particle  of  which  the 
mass  is  about  -§\-§  of  the  mass  of  a  hydrogen  atom.  Thus,  the 
cathode  rays  consist  of  electrons  which  are  thrown  off  from  the 
cathode  of  the  Crookes'  tube  at  high  velocity,  the  /3-rays  from  a 
radio-active  substance  such  as  uranium  are  electrons  which  are 
expelled  from  the  atoms  of  the  substance  at  high  velocity. 

A  simple  ion  is  an  atom  of  a  gas  from  which  a  negatively 
charged  electron  has  been  detached,  leaving  the  remainder  of  the 
atom  positively  charged.  Thus,  the  canal  rays  in  a  Crookes' 
tube  consist  of  simple  ions  positively  charged,  and  the  a-rays 
which  are  given  off  by  a  radio-active  substance  such  as  uranium 
consist  of  simple  ions  positively  charged.  A  compound  ion  con- 
sists of  a  negatively  charged  electron  or  a  positively  charged 
simple  ion  to  which  one  or  more  neutral  atoms  cling,  thus  form- 
ing a  charged  atomic  aggregate. 

lonization  by  the  electric  field. — According  to  the  kinetic 
theory  of  gases,  a  molecule  of  a  gas  travels  on  the  average  a 
certain  distance  between  successive  collisions  with  neighboring 
molecules.  This  distance  is  called  the  mean  free  path  of  the 
molecule.  The  mean  free  path  of  an  electron  in  a  gas  is  about 
4T/2J  times  as  great  as  the  mean  free  path  of  a  molecule  of  the 

*  See  J.  J.  Thomson,  Conduction  of  Electricity  Through  Gases,  page  II.' 
f  Called  a  corpuscle  by  J.  J.  Thomson. 
%  According  to  the  kinetic  theory. 


224        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

gas,  because  of  the  very  small  size  and  great  velocity  of  the 
electron,  whereas  the  mean  free  path  of  a  simple  or  compound 
ion  is  equal  to  or  even  less  than  .the  mean  free  path  of  a  molecule 
of  the  gas.  When  a  gas  is  subjected  to  an  electric  field  by 
being  placed  between  two  oppositely  charged  metal  plates,  a 
certain  amount  of  energy  is  imparted  by  the  electric  field  to  the 
electrons  between  successive  collisions,  and  a  much  smaller 
amount  of  energy  is  imparted  to  the  simple  or  compound  ions 
between  successive  collisions  (because  of  their  shorter  mean  free 
path).  If  the  energy  imparted  to  an  electron  between  successive 
collisions  exceeds  a  certain  value,  the  electron  is  able  to  ionize 
the  atoms  of  the  gas  when  it  collides  with  them,  producing  at 
each  collision  a  new  electron  and  a  simple  ion.  Similarly,  if 
the  energy  imparted  to  an  ion  between  successive  collisions  ex- 
ceeds a  certain  value,  the  ion  is  able  to  ionize  the  atoms  of  the 
gas  when  it  collides  with  them,  producing  at  each  collision  a 
new  ion  and  an  electron.  Thus,  the  electron  must  fall  freely 
through  a  certain  difference  of  potential  (about  30  volts)  in  order 
to  receive  enough  energy  to  ionize  air  molecules,  and  a  positive 
ion  must  fall  freely  through  a  certain  difference  of  potential  (about 
440  volts)  in  order  to  receive  enough  energy  to  ionize  air  molecules. 

127.  The  electric  spark  in  a  gas.  —  When  a  gas  is  subjected  to 
an  electric  field  of  which  the  intensity  is  sufficient  to  cause  both  * 
the  electrons  and  the  positive  ions  to  ionize  the  gas,  an  extremely 
rapid  increase  in  the  number  of  electrons  and  ions  takes  place, 
and  the  result  is  the  production  of  an  electric  spark.  The  mean 
free  path  of  the  positive  ions  in  a  gas  is  inversely  proportional  to 
the  pressure  of  the  gas  so  that  the  electric  strength  of  a  gas 

*  When  the  intensity  of  an  electric  field  is  sufficient  to  cause  only  the  electrons  to 
ionize  the  gas,  then  all  of  the  electrons  which  are  present  in  the  gas  flock  towards  the 
positive  electrode  forming  new  ions  and  new  electrons  on  the  way,  and  when  they 
reach  the  positive  electrode  the  action  ceases  except  for  the  occasional  formation  of 
an  electron  by  outside  influences.  When  the  electric  field  is  sufficiently  intense  to 
cause  electrons  and  positive  ions  both  to  produce  ionization,  then  new  ions  and  elec- 
trons are  formed  everywhere  between  the  electrodes  and  the  number  of  free  ions  and 
electrons  increases  indefinitely.  It  is  a  well-known  fact  that  an  electric  field  must 
continue  to  act  for  an  appreciable  time  before  a  spark  is  produced. 


THE   PHENOMENA   OF    ELECTROSTATICS.  22$ 

should  be  approximately  proportional  to  the  pressure.  This  is, 
in  fact,  the  case.  Thus,  the  dielectric  strength  of  air  at  normal 
atmospheric  pressure  is  about  20,000  volts  per  centimeter,  at  a 
pressure  of  10  atmospheres  the  strength  is  about  200,000  volts 
per  centimeter,  and  at  a  pressure  of  o.  I  atmosphere,  the  dielectric 
strength  is  about  2,000  volts  per  centimeter.  The  dielectric 
strength  of  air  reaches  a  minimum,  however,  at  a  pressure  of 
about  2  millimeters  of  mercury  and  increases  when  the  pressure 
falls  below  this  value.  An  electromotive  force,  sufficient  to  pro- 
duce a  spark  |  of  an  inch  long  in  air  at  atmospheric  pressure, 
will  produce  a  discharge  through  18  or  20  inches  of  air  at  2 
millimeters  pressure. 

The  idea  of  dielectric  strength  is  based  on  the  assumption  that 
the  electromotive  force  required  to  produce  a  discharge  is  propor- 
tional to  the  length  of  the  spark,  so  that  the  quotient,  volts  divided 
by  spark  length,  may  be  a  constant.  This  is  only  approximately 
true  in  gases  under  moderate  or  high  pressure,  and  when  the 
pressure  is  very  low  a  greater  electromotive  force  is  required  to 
strike  across  a  short  gap  than  is  required  to  strike  across  a  long 
gap.  This  curious  behavior  of  gas  at  low  pressure  is  illustrated 
by  a  famous  experiment  due  to  Hittorf.  Two  electrodes  were 
sealed  into  the  walls  of 
two  glass  bulbs  and  the 
tips  of  the  electrodes  were 
one  millimeter  apart,  as 
shown  in  Fig.  158.  The 
two  bulbs  were  connected 
together  by  a  spiral  tube 
375  centimeters  long,  and, 
when  the  pressure  of  the 

Fig.  158. 

gas  in  the  bulbs  was  re- 
duced to  a  very  low  value,  the  discharge  took  place  through 
the  long  tube  and  not  across  the  one  millimeter  gap  space  be- 
tween the  points  of  the  electrodes.* 

*  This  behavior  of  a  gas  at  low  pressure  is  fully  explained  by  the  atomic  theory. 
See  J.  J.  Thomson's  Conduction  of  Electricity  Through  Gases,  pages  430-527. 
16 


226        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


128.  The  Geissler  tube  and  the  Crookes  tube.  —  The  discharge 
of  electricity  through  ga$es  at  low  pressures  is  usually  studied  by 
means  of  a  glass  bulb  through  the  walls  of  which  are  sealed 
platinum  wires  which  terminate  'in  metal  plates  called  electrodes. 
The  current  enters  at  one  electrode,  the  anode,  and  passes  out  at 
the  other  electrode,  the  cathode.  This  bulb,,  which  is  called  a 
vacuum  tube,  is  filled  with  the  gas  to  be  studied  and  the  pressure 
is  reduced  to  any  desired  value  by  exhausting  the  tube  by  means 
of  an  air  pump. 

Before  exhaustion  the  discharge  through  the  tube  is  in  the  form 
of  a  sharply-defined  spark  similar  to  the  spark  in  the  open  air. 
When  the  pressure  of  the  gas  in  the  bulb  has  been 
reduced  to  a  few  centimeters  of  mercury,  the  spark 
begins  to  be  nebulous  and  a  continued  reduction  of 
pressure  causes  the  luminosity  ultimately  to  fill  the 
entire  tube.  When  the  pressure  has  been  reduced 
to  a  few  millimeters  of  mercury  the  discharge  pre- 
sents the  following  features,  as  shown  in  Fig.  159. 
There  is  a  thin  layer  of  luminosity  spread  over  the 
surface  of  the  cathode  C,  and  beyond  this  there 
is  a  comparatively  dark  space  D  called  the  Crookes 
dark  space,  the  width  of  which  depends  upon  the 
pressure  of  the  gas,  increasing  as  the  pressure  of 
the  gas  diminishes.  This  Crookes  dark  space  ex- 
tends to  a  boundary  which  is  approximately  a  sur- 
face traced  out  by  lines  of  constant  length  drawn 
normally  to  the  surface  of  the  cathode.  Beyond 
the  Crookes  dark  space  is  a  luminous  region  N 
called  the  negative  glow,  and  beyond  the  negative 
glow  is  another  comparatively  dark  region  F  which  is  called  the 
Faraday  dark  space.  Beyond  the  Faraday  dark  space  is  a  lumi- 
nous column  P  extending  to  the  anode  A  and  called  the  posi- 
tive column.  This  positive  column  usually  exhibits  alternations 
of  bright  and  dark  spaces  which  are  called  striations.  The  effects 
here  described  are  exhibited  at  their  best  in  a  vacuum  tube  in 


Fig.  159. 


THE    PHENOMENA   OF   ELECTROSTATICS.  22/ 

which  the  pressure  has  been  reduced  to  a  few  millimeters  of 
mercury.  Such  a  vacuum  tube  is  called  a  Geissler  tube.  When 
the  exhaustion  of  the  vacuum  tube  is  carried  further,  the  dark 
space  which  surrounds  the  cathode  (the  Crookes  dark  space)  ex- 
pands until  it  fills  the  entire  tube.  The  glass  walls  of  the  tube 
then  show  a  yellowish-green  or  blue  luminescence  (according  as 
the  tube  is  made  of  soda  glass  or  lead  glass)  and  a  slight  nega- 
tive glow  may  remain  in  portions  of  the  tube  remote  from  the 
cathode.  These  effects,  which  were  first  studied  by  Crookes  in 
England  and  by  Pliicker  and  Hittorf  in  Germany,  are  exhibited 
at  their  best  in  a  vacuum  tube  in  which  the  pressure  has  been 
reduced  to  a  few  thousandths  of  a  millimeter  of  mercury.  Such 
a  vacuum  tube  is  called  a  Crookes  tube. 

129.  Cathode  rays  and  canal  rays. — In  order  that  a  steady  dis- 
charge may  flow  through  a  vacuum  tube,  it  is  necessary  that  the 
electric  field  intensity  reach  a  value  sufficient  to  impart  to  the 
positive  ions  enough  energy  between  collisions  to  enable  them  to 
ionize  the  gas,  because  if  the  electrons  (negative  ions),  only,  pro- 
duce ionization,  the  discharge  through  the  tube  ceases  very  soon 
after  all  of  the  negative  ions  have  moved  across  to  the  neighbor- 
hood of  the  anode.  In  fact,  ionization  by  positive  ions  must  take 
place  in  the  neighborhood  of  the  cathode,*  and  it  is  this  necessity 
which  gives  rise  to  the  Crookes  dark  space.  The  action  which 
takes  place  in  the  Crookes  dark  space  is  as  follows  :  Electrons 
(negative  ions)  are  thrown  off  from  the  cathode  at  very  high 
velocity  by  the  intense  electric  field  in  the  Crookes  dark  space, 
very  energetic  ionization  takes  place  in  the  negative  glow  Nt 
Fig.  1 59,  and  the  positive  ions  that  are  produced  in  this  region 
attain  sufficient  velocity  in  traveling  towards  the  cathode  to  enable 
them  to  ionize  the  gas  in  the  immediate  neighborhood  of  the 
cathode.  That  is,  ionization  by  positive  ions  takes  place  in  the 
faint  glow  which  covers  the  cathode.  The  mutual  dependence 
of  the  ionization  which  takes  place  in  the  negative  glow  and  the 

.     *  A  detailed  discussion  of  this  matter  may  be  found  in  J.  J.  Thomson's  Conduction 
of  Electricity  Through  Gases,  pages  529-603. 


228        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

ionization  which  takes  place  in  the  faint  luminosity  in  the  imme- 
diate neighborhood  of  the  cathode  is  shown  by  placing  a  small 
obstacle  in  the  Crookes  dark  ..space.  This  obstacle  screens  a 
portion  of  the  cathode  surface  from  bombardment  by  the  positive 
ions  which  move  from  the  negative  glow  towards  the  cathode  so 
that  in  the  region  so  shaded  ionization  does  not  take  place.  In 
the  same  way  the  obstacle  also  screens  a  certain  portion  of  the 
negative  glow  from  bombardment  by  the  electrons  which  are 
thrown  from  the  cathode  and  this  portion  of  the  negative  glow 
ceases  to  exist  because  ionization  is  no  longer  produced  there. 
That  is  to  say,  the  obstacle  casts  a  shadow  on  the  cathode  and  it 
also  casts  a  shadow  into  the  negative  glow. 

The  electric  field  intensity  in  the  Crookes  dark  space,  being 
necessarily  sufficient  to  enable  the  positive  ions  to  produce 
ionization  at  the  surface  of  the  cathode,  is  able  to  impart  very 
much  greater  velocity  to  the  electrons  than  is  necessary  to  enable 
them  to  produce  ionization.  The  result  is  that  the  electrons 
which  are  thrown  off  from  the  cathode  travel  in  straight  lines 
through  a  long  portion  of  the  tube.  These  high  velocity  elec- 
trons constitute  what  are  called  cathode  rays.  The  cathode  rays 
are  faintly  visible  throughout  the  tube  because  of  occasional  col- 
lisions with  the  molecules 
of  the  gas. 

When  the  cathode  has 
a  small  hole  through  it, 
the  positive  ions  which 
move  towards  the  cathode 
from  the  negative  glow 
pass  through  this  hole  in 
the  form  of  a  stream  of 

Fig.  160.  i  •    i     •  i         •  ,.  , 

rays  which  is  made  visible 

by  the  luminosity  which  accompanies  the  collisions  of  the  posi- 
tive ions  with  the  molecules  of  the  gas.  Such  a  stream  of  posi- 
tive ions  constitutes  what  has  been  called  the  canal  rays. 

An  object  of  any  kind  placed  in  the  Crookes  tube  casts  a 
sharp  shadow  upon  the  wall  of  the  tube,  as  shown  in  Fig.  160. 


THE   PHENOMENA   OF   ELECTROSTATICS.  229 

The  wall  of  the  tube  shows  a  brilliant  luminescence  everywhere 
except  where  it  is  screened  by  the  obstacle  from  bombardment 
by  the  cathode  rays. 

Magnetic  deflection  of  cathode  rays  and  canal  rays.  —  A  mov- 
ing charged  body  is  equivalent  to  an  electric  current,  and  when  a 
charged  body  moves  across  a  magnetic  field  the  magnetic  field 
pushes  sidewise  upon  the  charged  body  and  causes  the  charged 
body  to  describe  a  curved  path.  The  magnetic  deflection  of  the 
cathode  rays  is  easily  shown  by  placing  a  horse-shoe  magnet 
with  its  poles  placed  on  opposite  sides  of  the  tube  shown  in  Fig. 
1 60.  The  shadow  of  the  cross  is  thrown  up  or  down  according 
to  the  arrangement  of  the  magnet.  The  magnetic  deflection  of 
the  canal  rays  is  very  slight ;  a  very  strong  magnetic  field  is 
necessary  to  produce  a  perceptible  deflection.  The  direction  of 
the  magnetic  deflection  of  the  cathode  rays  shows  that  these  rays 
are  negatively  charged  particles,  and  the  direction  of  the  mag- 
netic deflection  of  the  canal  rays  shows  that  these  rays  are  posi- 
tively charged  particles.  The  magnitude  of  the  deflection  of  the 
cathode  rays  shows  that  the  mass  of  the  cathode  particles  (elec- 
trons) is  very  small  and  that  their  velocity  is  very  great.  The 
magnitude  of  the  deflection  of  the  canal  rays  shows  that  the 
mass  of  the  canal  ray  particles  is  relatively  great  and  that  their 
velocity  is  less  than  the  velocity  of  the  cathode  rays.  This  mat- 
ter is  explained  in  detail  in  Art.  135. 

An  object  upon  which  the  cathode  rays  *  impinge  is  heated,  it 
may  be,  to  a  very  high  temperature.  Many  substances,  however, 
emit  light  (without  being  made  perceptibly  hot)  when  subjected 
to  bombardment  by  the  cathode  rays.  Such  substances  are  said 
to  be  luminescent.  For  example,  lead  sulphate  emits  a  deep 
violet  light,  zinc  sulphate  emits  white  light,  magnesium  sulphate, 
with  a  slight  admixture  of  manganese  sulphate,  emits  a  deep  red 
light  under  the  action  of  cathode  rays. 

*  The  cathode  rays  produce  effects  which  are  practically  important  and  which  can 
be  easily  observed.  The  effects  of  the  canal  rays,  however,  are  so  slight  as  to  be 
scarcely  perceptible  even  under  the  most  favorable  conditions.  Therefore  further  dis- 
cussion of  the  canal  rays  is  not  warranted  in  this  brief  outline. 


230        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  cathode  rays  pass  quite  readily  through  thin  metal  plates 
especially  through  thin  plates  of  aluminum.  By  using  a  Crookes 
tube  of  which  a  portion  of  the  wajj  is  made  of  thin  sheet  aluminum, 
the  cathode  rays  may  be  made  to  pass  through  into  the  outside 
air.  The  properties  of  cathode  rays  in  the  air  were  first  studied 
by  Lenard  who  found  that  they  produce  a  very  high  degree  of 
ionization  of  the  air  making  it  a  fairly  good  electrical  conductor. 
Lenard  found  the  cathode  rays  capable  of  traversing  from  10  to 
20  centimeters  of  air  at  atmospheric  pressure,  he  found  them 
capable  of  exciting  luminescence,  and  he  found  them  capable  of 
affecting  a  sensitive  photographic  plate. 

130.  The  Roentgen  rays.  —  Objects  upon  which  the  cathode 
rays  impinge,  not  only  become  heated  and  luminescent  as  de- 
scribed above,  but  they  emit  a  type  of  radiant  energy  which  was 
discovered  by  Roentgen  in  1 894.  Roentgen  rays  are  of  the  same 
physical  nature  as  light  rays,  that  is,  they  consist  of  waves  in  the 
ether,  and  they  are  related  to  light  waves  very  much  as  a  sharp 
"  razor"  wave  on  the  surface  of  water  would  be  related  to  a  long 
ocean  swell,  as  shown  in  Fig.  161.  Helmholtz  pointed  out  in 


"razor  wave1*  ocean  swell 

Fig.  161. 

1891  that  abrupt  wave  pulses  of  this  kind  in  the  ether  would  have 
certain  properties,  the  properties,  in  fact,  which  are  exhibited  by 
Roentgen  rays,  as  follows  :  These  rays  are  not  reflected  in  a  reg- 
ular way  by  the  surface  of  a  mirror,  and  they  are  not  refracted  by 
a  lens.  They  pass  through  all  substances,  subject  to  a  certain 
amount  of  absorption  which  is  greater  the  greater  the  density  of 
the  substance,  and  subject  to  a  certain  amount  of  diffused  scatter- 
ing. The  Roentgen  rays  affect  an  ordinary  photographic  plate 
and  they  have  a  powerful  ionizing  effect  on  gases. 

The  fluoroscope. —  Many  substances  such  as  barium  platinocy- 


Fig.    162. 


THE   PHENOMENA   OF   ELECTROSTATICS.  231 

anide  and  calcium  tungstate  become  luminescent  under  the  action 
of  Roentgen  rays.  This  effect  is  utilized  in  the  fluoroscope  which 
consists  of  a  cardboard  screen  covered  with  a  layer  of  barium 
platinocyanide.  When  the  Roentgen  ray  shadow  of  an  object, 
such  as  the  hand,  falls  on  this  screen  the  shadow  becomes  visible  ; 
where  the  Roentgen  rays  have  been  greatly  reduced  in  intensity 
by  the  bones  of  the  hand  the  screen  remains  dark,  where  the 
Roentgen  rays  have  been  slightly  reduced  in  intensity  by  the 
flesh  the  screen  is  moderately  luminous,  and  where  the  rays  have 
not  been  reduced  at  all  in  intensity  the  screen  is  highly  luminous. 
The  Roentgen  ray  shadow  of  an  object  may  be  rendered  visible 
by  allowing  it  to  fall  upon  a  photographic  plate  which  is  after- 
wards developed  like  an  ordinary  photographic  negative.  Thus, 
Fig.  162*  is  a  reproduction  of  a  shadow  photograph  of  a  wrist. 
The  focusing  tube.  —  In  order  that  a  shadow  may  be  sharply 
defined  the  radiation  which  produces  the  shadow  must  emanate 
from  a  very  small  source.  Figure  163  shows  a  Crookes  tube 


Fig.  163. 

with  a  concave  cathode  c  from  which  the  cathode  rays  converge 
and  strike  a  small  spot  on  a  platinum  plate  /.  This  small  spot  is 
the  source  of  the  Roentgen  rays.  Such  a  Crookes  tube  is  called 
a  focusing  tube,  and,  by  the  use  of  such  a  tube,  very  sharply 

*From  a  negative  by  Dr.  E.  W.  Caldwell,  President  (1908)  of  the  American 
Roentgen  Ray  Society. 


232        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

defined  Roentgen  ray  shadows  may  be  produced.  The  platinum 
plate  /  is  usually  connected  as  shown  to  the  aluminum  anode  a. 
An  interesting  feature  of  the  Crookes  tube,  which  is  shown  in  Fig. 
163,  is  the  small  platinum  tube  /  which  is  sealed  through  the 
glass  wall.  When  the  vacuum  in  the  Crookes  tube  becomes  too 
high  (presumably  by  the  transformation  of  the  ,residual  gases  into 
non-volatile  products),  the  small  tube  t  is  held  for  a  few  seconds 
in  the  flame  of  an  alcohol  lamp  and  a  sufficient  amount  of  hydro- 
gen passes  through  the  hot  platinum  to  replenish  the  supply  of 
gas  in  the  Crookes  tube. 

131.  Conductivity  of  hot  gases  and  flames.  —  A  hot  gas  is  a 
fairly  good  electrical  conductor  and  this  conductivity  has  been 
found  to  be  due  to  the  presence  of  free  ions.*     The  conductivity 
of  a  hot  gas  or  flame  is  shown  by  the  fact  that  a  charged  glass 
rod  may  be  completely  discharged  by  passing  the  flame  of  a  Bun- 
sen  burner  rapidly  over  its  surface. 

132.  The  electric  arc.  —  In  order  to  produce  a  perceptible  dis- 
charge of  electricity  (flow  of  current)  through  a  gas,  a  very  high 
electromotive  force  must  be  used  because  of  the  necessity  of  pro- 
ducing ionization  in  the  gas  by  the  collision  of  the  moving  ions 
with  the  gas  molecules  ;  and  the  amount  of  current  which  can  be 
made  to  flow  through  a  gas  is  usually  very  small  because  of  the 
comparatively  small  number  of  these  ions.     When,  however,  metal 
or    carbon    electrodes    are  heated   to   a  very  high   temperature 
they  emit   electrons  (negative   ions)  in  great    numbers  f    and  a 
very  considerable  current  may  then  be  made  to  flow  through  the 
intervening   gas.     Thus,  a  current  of  an   ampere  or  more  may 
be  made  to  flow  between  a  cold  metal  anode  and  a  very  hot  metal 
cathode  in  a  vacuum  tube.     When  two  carbon  rods  are  connected 
to  a  battery  or  dynamo,  brought  into  contact  and  then  separated, 
the  current  which  begins  to  flow  across  the  indefinitely  small  gap 
between  the  two  carbon  rods  raises  the  tips  of  the  carbons  to  a 

*  See  J.  J.  Thomson's  Conduction  of  Electricity  Through  Gases,  pp.  228—249. 
7  See  J.  J.  Thomson's  Conduction  of  Electricity  Through  Gases,  pp.  188-227. 


THE    PHENOMENA   OF   ELECTROSTATICS.  233 

very  high  temperature  so  that  electrons  (negative  ions)  are  emitted 
in  great  numbers.  The  result  is  that  the  current  continues  to 
flow  between  the  carbon  tips.  The  column  of  hot  vapor  between 
the  carbon  tips  is  called  an  electric 
arc,  and  the  intense  heating  of  the 
two  carbon  tips  is  due  to  their  bom- 
bardment by  the  ions  which  move 
across  the  arc  and  carry  the  electric 
current.  The  electric  arc  may  be 
easily  maintained  between  a  hot  neg- 
ative carbon  (cathode)  and  a  rapidly 
rotating  disk  (a  cold  anode),  but  not 
between  a  cold  cathode  and  a  hot 
anode.  This  shows  that  the  emis- 
sion of  negative  ions  (electrons)  by 
the  hot  carbon  is  essential  to  the 
formation  of  the  electric  arc.  The 
appearance  of  the  arc  between  car- 
bon electrodes  is  shown  in  Fig. 
164.* 

133.  Chemical  effect  of  the  dis- 
charge through  gases.  —  The  discharge  of  electricity  through 
gases  is  accomplished  by  the  ionization  of  the  gas  as  above  ex- 
plained. This  ionization  means  not  only  the  breaking  down  of 
the  molecules  of  a  compound  gas  but  also  the  separation  of  elec- 
trons from  the  individual  atoms  of  the  constituents  of  the  com- 
pound gas.  The  ionization  of  mixed  gases  promotes  chemical 
combination.  Thus,  the  nitrogen  and  oxygen  of  the  air  combine 
slowly  under  the  action  of  the  electric  spark. 

When  oxygen  (or  air)  is  ionized,  the  recombination  of  the 
oxygen  ions  results  in  the  production  of  ozone.  Thus  the 

*  The  properties  of  the  electric  arc  are  discussed  in  great  detail  in  a  paper  by  C. 
P.  Steinmetz,  Trans,  International  Electrical  Congress,  Vol.  II,  pages  710-730,  St. 
Louis,  1904  ;  in  a  paper  by  W.  R.  Whitney,  Trans.  American  Electrochemical 
Society,  Vol.  7,  pages  291-299,  1905  ;  and  in  J.  J.  Thomson's  Conduction  of  Elec- 
tricity Through  Gases,  pages  604-620. 


234        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

peculiar  odor  which  is  given  off  by  a  Toepler-Holtz  machine  or  a 
Wimshurst  machine  is  due  to  the  ozone  which  is  formed.  The 
action  which  takes  place  in  the  formation  of  ozone  from  oxygen 
is  as  follows :  Ordinary  oxygen  is  bi-atomic,  that  is,  it  contains 
two  atoms  of  oxygen  in  the  molecule.  lonization  causes  the  dis- 
integration of  these  bi-atomic  molecules  forming  mono-atomic 
oxygen,  and  this  mono-atomic  oxygen  recombines  forming  a 
large  proportion  of  bi-atomic  oxygen  again  and  a  small  propor- 
tion of  tri-atomic  oxygen,  or  ozone.  In  the  production  of  ozone 
for  commercial  purposes  a  blast  of  air  is  driven  between  two 
metal  plates  which  are  connected  to  a  high  voltage  alternator. 
The  repeated  reversals  of  the  high  electromotive  force  between 
the  plates  ionizes  the  intervening  air  repeatedly,  and  the  recom- 
bination of  the  ions  is  accompanied  by  the  formation  of  a  certain 
percentage  of  ozone,  as  above  explained.  In  order  to  produce  a 
high  degree  of  ionization  throughout  the  entire  region  between 
the  two  metal  plates,  it  is  necessary  to  place  a  thin  plate  of  glass 
between  the  metal  plates  so  as  to  prevent  the  formation  of  a 
single  spark  from  plate  to  plate.  The  effect  of  this  glass  plate  is 
to  cause  a  fine  brush  discharge  to  take  place  throughout  the 
entire  region.  Without  the  glass  plate  a  single  brilliant  spark 
passes  through  the  air.  With  the  glass  plate,  a  diffused  violet 
luminosity  is  produced  throughout  the  region  between  the  metal 
plates. 

134.  Radio-activity.*  —  The  chemical  elements  uranium,  tho- 
rium, and  radium  and  their  compounds  have  the  property  of 
making  a  surrounding  gas  an  electrical  conductor.  Thus,  one 
ten-millionth  of  a  gram  of  radium  bromide  which  is  left  as  a 
residue  upon  a  metal  plate  by  evaporating  a  small  quantity  of  a 
dilute  solution  of  radium  bromide  on  the  plate,  causes  a  gold 
leaf  electroscope  to  be  discharged  in  a  few  seconds  when  the 

*The  student  is  referred  to  the  following  books  for  a  full  discussion  of  radio- 
activity :  Radioactivity,  by  E.  Rutherford,  Cambridge,  1905  (second  edition); 
Radioactivity,  by  Frederick  Soddy,  London,  1904  ;  and  Radioactive  Transforma~ 
tions,  by  E.  Rutherford,  New  York,  1906. 


THE   PHENOMENA   OF   ELECTROSTATICS.  235 

radium -covered  plate  is  held  near  to  the  metal  plate  of  the  elec- 
troscope. Uranium  and  thorium  have  the  same  effect  but  the 
discharge  which  they  produce  is  not  so  rapid  unless  a  large 
quantity  of  material  is  employed.  This  property  of  these  metals 
and  of  their  compounds  is  called  radio-activity,  a  name  which 
originated  because  of  the  peculiar  radiations  which  are  given  off 
by  radio-active  substances  and  to  which  the  discharging  action  is 
due.  These  radiations  are  of  three  distinct  kinds,  which  are 
called  the  a-rays,  the  /3-rays,  and  the  7-rays,  respectively.  The 
7-rays  penetrate  through  a  foot  or  more  of  solid  metal  or  through 
many  feet  of  air ;  the  /3-rays  penetrate  through  a  moderate  thick- 
ness of  a  light  metal,  such  as  aluminum  ;  whereas  the  a-rays  are 
stopped  by  a  very  thin  layer  of  aluminum  or  by  a  layer  of  air 
two  or  three  inches  in  thickness. 

The  a-rays  consist  of  positive  ions  each  about  twice  as  massive 
as  a  hydrogen  atom.  These  ions  are  projected  from  the  radio- 
active substance  at  a  velocity  of  about  20,000  miles  per  second, 
and  each  of  them  ionizes  about  100,000  air  molecules  before  it  is 
brought  to  rest  by  repeated  collision  After  traveling  two  or 
three  inches  through  the  air,  the  velocity  of  these  a-particles  is 
reduced  to  so  low  a  value  as  to  render  them  no  longer  perceptible 
by  their  ionizing  effects. 

The  /3-rays  consist  of  electrons  (negative  ions)  each  about  -g-J-g- 
as  massive  as  a  hydrogen  atom.  These  electrons  are  projected 
from  the  radio-active  substance  at  a  velocity  which  in  some  cases 
is  nearly  as  great  as  the  velocity  of  light  (186,000  miles  per 
second).  The  /8-particles  also  have  the  property  of  ionizing 
the  gas  through  which  they  pass  but  not  to  so  great  an  extent 
as  the  a-particles,  and  they  travel  several  feet  through  the  air 
before  their  velocity  is  reduced  to  so  low  a  value  as  to  render 
them  no  longer  perceptible  by  their  ionizing  effects. 

The  7-rays  are  extremely  abrupt  waves  in  the  ether  essentially 
the  same  in  character  as  Roentgen  rays,  but  much  more  penetrat- 
ing than  ordinary  Roentgen  rays.  The  7-rays  also  have  the 
property  of  ionizing  a  gas. 


236        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  a-rays  and  the  /3-rays  are  deflected  by  the  magnetic  field 
and  by  the  electric  field.  The  direction  of  the  deflection  of  the 
a-rays  is  in  each  case  opposite  to  the  direction  of  deflection  of  the 
/3-rays,  and  therefore  it  is  known  that  the  a-particles  are  positively 
charged  and  that  the  y3-particles  are  negatively  charged.  The 
7-rays  are  not  deflected  by  a  magnetic  field,  or  by  an  electric 
field. 

The  present  hypothesis  regarding  radio-activity  is  that  the  atoms 
of  all  substances  are  complex  systems  of  excessively  small  particles 
called  electrons,  the  atom  of  each  element  being  a  characteristic 
self-contained  group  or  system  of  electrons  in  very  violent  orbital 
motion.  These  systems  of  electrons  (atoms)  are  supposed  to  be 
to  some  extent  unstable,  and  when  instability  occurs,  the  system 
(atom)  collapses  into  a  new  configuration  and  at  the  same  time 
expels  one  or  more  positively  or  negatively  charged  electrons  or 
groups  of  electrons  which  constitute  the  a-rays  and  the  yS-rays. 
According  to  this  hypothesis  the  7-rays  consist  of  abrupt  ether 
waves  which  are  produced  by  the  sudden  collapse  of  the  atomic 
structure  when  instability  occurs.* 

A  clear  representation  of  the  nature  of  a.-,  fi-,  and  7-rays  is 
shown  in  Fig.  165.  Imagine  an  atom  of  the  radio-active  material 
to  collapse  at  a  given  instant  sending  out  a  7-wave,  an  a-particle, 
and  a  /3-particle.  The  relative  positions  reached  by  the  7-wave, 
the  a-particle,  and  the  /3-particle  at  a  given  instant  are  shown  in 
the  figure.  The  a-particle  is  a  large  positively  charged  particle 
and  the  /3-particle  is  a  small  negatively  charged  particle. 

135.  Determination  of  velocity  and  mass  of  the  particles  which 
constitute  canal  rays  (or  a-rays)  and  cathode  rays  (or  /3-rays).  — 
A  narrow  stream  of  rays  from  a  radio-active  substance  may  be 

*  A  very  instructive  discussion  of  the  electron  theory  is  given  by  Sir  Oliver  Lodge 
in  a  book  entitled  Electrons,  published  by  Geo.  Bell  &  Sons,  London,  1906.  The 
method  of  measuring  the  degree  of  radio-activity  of  a  radio-active  substance  is  explained 
in  Franklin,  Crawford  and  MacNutt's  Practical  Physics,  Vol.  2,  pages  148-153. 
An  example  of  the  study  of  a  radio-active  transformation,  that  is,  of  the  change  which 
takes  place  in  the  radio-active  substance,  as  a  result  of  its  radio-activity,  is  given  in 
Franklin,  Crawford  and  MacNutt's  Practical  Physics,  Vol.  2,  pages  154  and  155. 


THE   PHENOMENA   OF    ELECTROSTATICS. 


237 


obtained  by  the  arrangement  shown  in  Fig.  166  in  which   AB  is 
a  sensitive  photographic  plate  upon  which  the  narrow  stream  of 


photographic  plate 


\   narrow  stream 
Y        of  rays 


^-radio-active 
material 


^*  radio-active 
material1 


Fig.  165. 


Fig.  166. 


rays  impinges.  Figure  1 67  shows  the  effect  of  an  electrical  field 
upon  a  thin  stream  of  rays  from  a  radio-active  substance.  The 
direction  of  the  electric  field  is  shown  by  the  fine  horizontal  arrows 

A  photographic  plate1 7? 


electrically 
charged 
plate      ^ 

(positive)     I 


_  electrically 

charged 
^      plate 

(negative") 


/////////////////////I 
lead  block 

J///////////////M. 


lead  block 
'///////////////ft/ft 


material 


Fig.  167. 


(the  lines  of  force  of  the  electric  field  pass  from  the  positively 
charged  plate  to  the  negatively  charged  plate).     The  effect  of  the 


238        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


electrical  field  is  to  deflect  the  a-particles  in  the  direction  of  the 
field  and  the  /3-particles  in  the  opposite  direction,  while  the  7-rays 
are  not  affected  at  all.  -  The  amoynt  of  deflection  in  each  case  may 

A  photographic  plate         B 


jV-poJe  of 
magnet 


lead  block 


'/        lead  block 


iue  material 

Fig.  168. 

a-particles  deflected  towards  the  reader. 
/3-particles  deflected  away  from  the  reader, 
y-waves  not  deflected  at  all. 

be  determined  by  developing  the  photographic  plate  upon  which 
the  rays  impinge.  The  effect  of  the  magnetic  field  upon  the  rays 
from  a  radio-active  substance  is  shown  in  Fig.  168  in  which  the 
fine  horizontal  lines  represent  the  lines  of  force  of  a  magnetic  field 
between  the  two  large  magnet  poles. 

The  determination  of  the  velocity  of  the   a-   and    yS-particles 
is  somewhat  analogous  to  the  following  method  for  determining 


Fig.  169. 

the  velocity  of  a  cannon  ball.  The  curved  line  in  Fig.  169 
represents  the  orbit  of  a  cannon  ball,  D  being  the  horizontal 
distance  traveled  by  the  ball  in  a  given  time  and  d  being  the 


THE   PHENOMENA   OF   ELECTROSTATICS. 


239 


vertical  distance  fallen  by  the  ball  under  the  action  of  gravity. 
If  D  is  known  and  d  observed,  then  the  velocity  of  the  cannon 
ball  is  given  by  the  equation 


(0 


2d 


in  which  g  is  the  acceleration  of  gravity. 

Action  of  the  electrical  field  on  a  moving  charged  particle.  — 
Consider  a  charged  particle  moving  upwards  through  an  elec- 
trical field  as  shown  in  Fig.  170.  Let  q  be  the  charge  on  the 


lead  block  \  V  lead  bfa£ 


lead  block 


had  block 


Fig.  170.  Fig.  171. 

Magnetic  field  perpendicular  to  plane  of  paper. 

particle  in  abcoulombs  and  e  the  intensity  of  the  electrical  field 
in  abvolts  per  centimeter.  Then  the  force  F  in  dynes  pulling 
on  the  particle  is  equal  to  qe,  so  that  the  acceleration  of  the 
particle  in  the  direction  of  F  is  qejm.  It  is  evident  that  the 
particle  moves  in  the  same  sort  of  an  orbit  as  a  cannon  ball,  and 
that  the  acceleration  qejm  corresponds  to  the  acceleration  of 
gravity  g  in  the  case  of  a  cannon  ball.  Therefore,  using  qejm 
for  g  in  equation  (i),  we  have 


or 


~  2dm 
m       D*e 


(iii) 


240        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Action  of  the  magnetic  field  on  a  moving  charged  particle.  — 
Figure  171  represents  a  charged  particle  moving  upwards  through 
a  magnetic  field,  the  lines  of  force  of  which  are  perpendicular  to 
the  plane  of  the  figure.  The  moving  particle  is  equivalent  to  an 
electric  current,  and  the  side  force  F  is  equal  to  qvh  where  q 
is  the  charge  on  the  particle  in  abcoulombs, ,  v  is  its  velocity  in 
centimeters  per  second,  and  //  is  the  intensity  of  the  magnetic 
field  in  gausses.  Therefore  the  acceleration  of  the  particle  in  the 
direction  of  F  is  qvhjm.  The  force  F  is  continuously  at  right 
angles  to  v  so  that  the  particle  describes  a  circular  orbit.  But 
the  acceleration  of  a  particle  moving  in  a  circular  orbit  is  v2  jr, 
and  the  relation  between  the  radius  of  the  circle  r,  the  semi- 
chord  D,  and  the  versed  sine  d  is 

D2 

r=2d 
Therefore  we  have 

qvh 
~^ 
whence 

m 

—  =  — -  -  (iv) 

q       2dv 

Determination  of  velocity  of  particles.  —  Reduced  to  the  simplest 
terms,  the  method  of  determining  velocity  may  be  described  as 
follows  :  An  electrical  field  e  in  the  plane  of  the  paper,  Fig. 
170,  and  a  magnetic  field  h  at  right  angles  to  the  plane  of  the 
paper  in  Fig.  170  are  adjusted  so  that  together  they  produce  no 
deflection  of  the  particles  which  are  being  studied.  When  this 
condition  is  realized,  the  force  qe  with  which  the  electrical  field 
acts  on  the  moving  particles  is  equal  and  opposite  to  the  force 
qvh  with  which  the  magnetic  field  acts  on  the  moving  particles, 
so  that,  disregarding  sign,  we  have 

qe  =  qvh 
or 


THE   PHENOMENA   OF   ELECTROSTATICS.  241 

that  is  to  say,  the  velocity  of  the  particles  is  equal  to  the  ratio 
of  the  electric  field  intensity  e  in  abvolts  per  centimeter  to  the 
magnetic  field  intensity  h  in  gausses,  on  the  condition  that  the 
combined  action  of  the  fields  produces  no  deflection  of  the  mov- 
ing particles. 

"Electrochemical  equivalent"  of  a-  and  ft- par  tides. — Ac- 
cording to  the  dissociation  theory  of  electrolysis  each  atom  of 
hydrogen,  for  example,  in  a  dilute  solution  of  sulphuric  acid  is 
isolated  and  carries  a  definite  amount  of  charge,  and  the  ratio 
(w/<7)  of  the  mass  m  of  a  hydrogen  atom  (ion)  to  the  charge 
q  upon  it  is  equal  to  the  electrochemical  equivalent  of  hydrogen, 
or,  in  other  words,  to  the  number  of  grams  of  hydrogen  which 
are  liberated  during  the  passage  of  one  coulomb  of  electric  charge 
through  an  electrolytic  cell  containing  dilute  sulphuric  acid.  The 
ratio  (jnjq)  of  the  mass  of  a  gas  ion  to  the  charge  upon  the  ion 
is  called  the  "electrochemical  equivalent"  of  the  gas  ion.  This 
ratio  is  determined  by  equation  (iii)  or  (iv)  when  the  electric  or 
magnetic  deflection  of  the  particle  has  been  observed  and  when 
the  velocity  of  the  particle  is  known.  The  value  so  determined 
is  given  in  grams  per  abcoulomb  and  it  is  equal  to  5.36  x  io~8 
grams  per  abcoulomb  for  the  /3-particles  (electrons),  from  which 
it  follows  that  the  particles  have  a  mass  -g-^-  as  great  as  the  mass 
of  a  hydrogen  atom  if  the  charge  q  is  the  same  in  both  cases.* 

*  In  regard  to  the  equality  of  charge  on  mono-valent  ions  in  electrolytes  and  on  gas 
ions,  see  Oliver  Lodge's  Electrons,  pages  77-90,  where  a  simple  account  is  given  of 
the  work  which  has  been  done  by  J.  J.  Thomson  in  the  determination  of  the  value  of 
q  (or  /»). 


17 


CHAPTER   IX. 
ELECTRIC  OSCILLATIONS  AND  ELECTRIC  WAVES. 

136.  Mechanical  conceptions  of  the  magnetic  and  electric  fields.* 

—  The  foregoing  chapters  are  devoted  to  the  discussion  of  the 
phenomena  of  the  electric  current  and  the  phenomena  exhibited  by 
electrically  charged  bodies.  The  phenomena  of  electric  oscilla- 
tions and  especially  the  phenomena  of  electric  waves  have  not 
as  yet  been  touched  upon.  It  is  usual  to  treat  these  phenomena 
on  the  basis  of  the  differential  equations  of  the  electro-magnetic 
field,  but  it  is  needless  to  say  that  this  mode  of  treatment  cannot 
be  followed  in  an  elementary  text.  The  most  satisfactory  ele- 
mentary treatment  of  electric  oscillations  and  electric  waves  is  to 
develop  the  mechanical  conceptions  of  the  magnetic  and  electric 
fields  and  thus  arrive  at  a  rational  insight  into  electro-magnetic 
phenomena.  This  method  is  followed  in  this  chapter. 

Maxwell  was  the  first  to  work  out  mechanical  conceptions  of 
magnetic  and  electric  fields,  and  Maxwell's  conceptions  are  used 
in  the  present  chapter  f  although  certain  inconsistencies  arise  in 
the  attempt  to  extend  these  conceptions  to  three  dimensions. 

*Sir  Oliver  Lodge's  Modern  Views  of  Electricity  is  perhaps  the  best  elementary- 
treatise  on  this  subject.  This  book  is  now  (1908)  being  rewritten. 

tThe  most  complete  mechanical  conception  of  the  electro-magnetic  field  is  that 
which  is  based  upon  Lord  Kelvin's  gyrostatic  model  of  the  ether.  This  gyrostatic 
model  of  the  ether  is  a  mechanical  structure  which  is  capable  of  reproducing  most  of 
the  known  phenomena  of  electricity  and  magnetism  and  of  light.  See  SEther  and 
Matter,  by  Joseph  Larmor,  Appendix  F-,  Cambridge,  1900.  Lord  Kelvin's  gyrostatic 
model  of  the  ether  has  led  to  a  hydrodynamic  conception  of  the  ether,  due  chiefly  to 
Larmor,  in  which  the  ether  is  assumed  to  be  a  perfect  fluid  which  is  endowed  with 
the  necessary  elastic  properties  by  an  indefinitely  fine  grained  whirling  motion.  On 
the  basis  of  Lord  Kelvin's  gyrostatic  conception  of  the  ether  and  also  on  the  basis  of 
Larmor' s  turbulent  ether,  the  magnetic  field  is  thought  to  consist  of  a  simple  flow  of 
the  ether  along  the  lines  of  force  of  the  magnetic  field.  This  conception  of  the 
magnetic  field  is  very  different  from  the  conception  which  is  outlined  in  this  text  and 
which  is  based  upon  Maxwell's  conception  of  the  ether. 

242 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     243 


Fig.  172. 


137.  Maxwell's  mechanical  model  of  the  ether.  —  The  ether  is 
to  be  considered  as  built  up  of  very  small  cells  of  two  kinds,  posi- 
tive and  negative,  in  such  a  way  that  only  unlike  cells  are  in 
contact.  These  cells  are  imagined  to  be  gear  wheels  provided 
with  rubber-like  teeth,  as  shown  in  Fig. 
172,  so  that  if  a  cell  be  turned  while  the 
adjacent  cells  are  kept  stationary,  then  a 
torque  due  to  elastic  distortion  of  the  gear 
teeth  is  brought  to  bear  upon  the  turned 
cell.  In  subsequent  figures,  these  cells  or 
cog-wheels  are  represented  by  plain  circles 
for  the  sake  of  simplicity. 

Conception  of  the  magnetic  field.  — The 
ether  cells  at  a  point  in  the  magnetic  field  are  thought  of  as  ro- 
tating about  axes  which  are  parallel  to  the  direction  of  the  field 
at  the  point,  the  angular  velocity  of  the  cells  being  proportional 
to  the  intensity  of  the  field.  The  positive  cells  rotate  in  the  di- 
rection in  which  a  right-handed  screw  would  be  turned  that  it 
might  move  in  the  direction  of  the  field, 

)     ("h)     (i)     anc*  ^e  neSa^ve  ceUs  rotate  in  the  oppo- 
(^\     (T\         site  direction.     This  opposite  rotation  of 
)     (Q     Q)    positive  and  negative  cells   is  mechanic- 
(^)     (7?)         ally  possible  since  only  unlike  cells  are 
ff)     ©     &}     £3    geared  together.     This    rotatory   motion 
(3r    t^    fe^<        °f  ^e  etner  ceUs  is   shown  in   Fig.  173, 
which  represents  a  magnetic  field  perpen- 
dicular to  the  plane    of  the  paper  and 

directed  away  from  the  reader ;  all  the  positive  cells  are  rotating 
clockwise  and  all  the  negative  cells  are  rotating  counter-clock- 
wise. The  energy  of  the  magnetic  field  (see  Art.  44)  is  repre- 
sented by  the  kinetic  energy  of  rotation  of  the  ether  cells. 

Conception  of  the  electric  field. — The  positive  ether  cells  at  a 

point  in  an  electric  field  are  thought  of  as  being  displaced  in  the 

;    direction  of  the  field,  while  the  negative  cells  are  displaced  in  the 

-•    opposite  direction,  and  this  displacement  is  assumed  to  be  pro- 


244        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

portional  to  the  electric  field  intensity.     Thus,  Fig.   174  repre- 
sents the   case  in  which  the  positive  cells  have  been  displaced 
towards  the  bottom  of  ,the  page  relatively  to  the  negative  cells  as 
shown    by   the   arrows,   that  :is    to    say,   the 
distortion  of  the   ether  structure  in  Fig.  174 
represents    an    electrjc 
J          j     *     J      field    directed    toward 


ft)  '  ft)  '  ft)  '  ft)  the  bottom  of  the  page, 
t/     t/      0         Figure   175    represents 

meshes  of  the  cell-  / 


So/         ular  structure    of    the 
(S^©^©X~N(^    ether.      These   two 

{~/     \~J     \-=J         meshes  are    square   in 

Fig.  174.  the   undistorted    ether,  F[z-  175« 

as  shown  in  Fig.  173,  whereas  the  downward  displacement  of 
the  positive  cells  in  Fig.  174  has  distorted  these  meshes,  as  shown 
in  Figs.  174  and  175.  Inasmuch  as  the  cell  structure  of  the 
ether  is  assumed  to  be  elastic  (the  gear  teeth  in  Fig.  172  being 
made  of  a  substance  like  rubber),  the  distortion  of  the  ether 
structure  which  is  shown  in  Fig.  1 74  represents  potential  energy 
and  this  energy  is  the  energy  of  the  electric  field  (see  Art.  104). 

Nearly  the  whole  of  the  following  discussion  is  based  upon  the 
torque  action  which  is  exerted  upon  a  given  cell  by  the  elastic 
distortion  which  is  represented  in  Fig.  175.  77«>  torque  action  is 
the  connecting  link  between  the  electric  field  and  the  magnetic  field 
and  a  clear  understanding  of  it  is  of  the  utmost  importance.  Con- 
sider the  two  positive  cells  to  the  right  of  the  middle  cell  in  Fig. 
175.  Inasmuch  as  these  two  positive  cells  have  been  displaced 
downwards  with  respect  to  the  middle  cell,  they  exert  torques 
upon  the  middle  cell  as  shown  by  the  arrows  c  and  d,  and 
these  torques  are  proportional  to  the  intensity  of  the  electric  field, 
that  is,  to  the  downward  displacements  of  the  cells.  The  two 
positive  cells  to  the  left  of  the  middle  cell  in  Fig.  175  exert 
torques  which  are  equal  to  c  and  d  respectively,  but  opposite 
in  direction. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     245 

138.  The   energy  stream    in    the    electromagnetic    field.  —  A 

region  in  which  electric  field  and  magnetic  field  co-exist  may  be 
called  an  electromagnetic  field  for  the  sake  of  brevity.  It  has 
been  shown  by  J.  H.  Poynting  *  from  theoretical  considerations 
that  energy  streams  through  an  electromagnetic  field  in  a  direction 
which  is  at  right  angles  both  to  the  electric  field  and  to  the  mag- 
netic field  at  each  point,  and  that  the  amount  of  energy  per  second 
which  streams  across  one  square  centimeter  of  area  is  propor- 
tional to  the  product  of  the  electric  and  magnetic  field  intensities. 
In  case  the  electric  and  magnetic  fields  are  not  at  right  angles  to 
each  other,  the  energy  stream  is  proportional  to  the  product  of 
the  intensities  of  the  two  fields  and  the  sine  of  the  included  angle. 
Conception  of  the  energy  stream.  —  Consider  a  row  of  gear 
wheels  as  shown  in  Fig.  176.  Imagine  the  wheel  W  to  be 


frank 


Fig.   176. 


turned  steadily  by  a  crank,  and  the  wheel  Wf  to  be  hindered 
by  a  brake.  The  result  is  that  energy  is  continuously  trans- 
mitted along  the  chain  of  gear  wheels  from  W  to  W1 ',  any 
given  gear  of  the  chain  is  acted  upon  by  equal  and  opposite 
torques  by  the  gear  wheels  on  each  side  of  it,  the  transmission 
of  energy  by  the  chain  depends  upon  this  torque  action  combined 
with  the  motion  of  the  wheels,  and  the  rate  at  which  energy  is 
transmitted  along  the  chain  is  proportional  to  the  product  of  the 
speed  of  the  wheels  and  the  torque  action  between  adjacent  wheels. 
Imagine  the  ether  cells  in  Fig.  1 74  to  be  rotating,  positive  cells 
in  one  direction,  negative  cells  in  the  other,  about  axes  perpen- 
dicular to  the  plane  of  the  paper.  This  rotatory  motion  consti- 
tutes a  magnetic  field  perpendicular  to  the  plane  of  the  paper  and 
perpendicular  to  the  electric  field  which  is  towards  the  bottom  of 

*See  Philosophical  Transactions,  Vol.  175,  Part  II,  page  343,  1884. 


246         ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

the  page.  On  account  of  the  torque  action  between  the  cells,  as 
explained  in  connection  with  Fig.  175,  energy  will  be  transferred 
to  the  right  (or  left)  by.  each  horizontal  chain  of  geared  cells  at  a 
rate  which  is  proportional  to  the  product  of  the  intensity  of  the 
magnetic  field  and  the  intensity  of  the  electric  field  ;  and  the  energy 
per  second  flowing  across  an  area  (of  which  the  normal  is  perpen- 
dicular to  both  electric  field  and  magnetic  field)  is  proportional  to 
the  product  of  the  respective  field  intensities  and  proportional  to 
the  area,  inasmuch  as  the  area  is  proportional  to  the  number  of 
rows  of  cells  which  are  acting  as  chains  of  gear  wheels.  There- 
fore the  energy  stream,  that  is,  energy  per  unit  area  per  second, 
is  proportional  to  the  product  of  magnetic  and  electric  field  inten- 
sities and  it  is  at  right  angles  to  both. 

139.  The  electric  current. —  Consider  a  wire  AB,  Fig.  177, 
along  which  an  electric  current  is  flowing  from  B  towards  A. 

The  magnetic  field  on  oppo- 
site sides  of  AB  is  in  opposite 
directions,  so  that  the  positive 
ether   cells  at  /  and  /'  are 
"      rotating  in  opposite  directions 
as  shown.     An  electric   cur- 
rent   may  be   maintained   for 
an  indefinite  length   of  time, 
FIg:- 177-  but   the    opposite  rotation  of 

positive  ether  cells  on  the  two  sides  of  AB,  Fig.  177,  cannot 
be  accommodated  by  an  ever-increasing  ether  distortion  (distor- 
tion of  the  rubber-like  teeth  of  the  ether  cells  as  shown  in  Fig. 
172),  there  must  be  a  slip  between  adjacent  cells  somewhere  be- 
tween/ and  p' .  This  slip  between  adjacent  ether  cells  takes  place 
in  the  material  of  the  wire  and  constitutes  an  electric  current. 

Steady  electric  currents  flow  in  closed  circuits. —  Let  AB,  Fig. 
178,  be  a  wire  *  in  which  a  steady  electric  current  is  flowing  from 

*  In  Fig.  178,  as  in  all  subsequent  figures,  a  wire  is  to  be  thought  of  as  an  indefi- 
nitely broad  metal  sheet,  because  the  cellular  conception  of  the  ether  is  not  adapted  to 
three  dimensions. 


ELECTRIC    OSCILLATIONS   AND    ELECTRIC   WAVES.     247 


Fig.  178. 


B  towards  A.  Consider  the  opposite  rotation  of  like  ether  cells 
at  /  and  /',  and  consider  a  chain  of  geared  cells  passing  from 
p  to  pf  around  the  end  of  AB.  The  current  through  AB 
may  flow  for  an  indefinite  time  and  therefore  the  opposite  rota- 
tion of  the  positive  ether  cells 
at  p  and/'  may  continue  in-  PI 

definitely,  but  this  continued 
opposite  rotation  at  /  and  /' 
cannot  be  accommodated  by 
an  ever-increasing  distortion 
of  the  elastic  gear  teeth  of 
the  ether  cells  along  the  chain 
of  geared  cells  which  pass 
around  the  end  of  AB.  A  slip  must  take  place  between  ad- 
jacent cells  at  some  point  along  this  chain.  Therefore  the  line 
of  flow  of  the  current  AB  (line  of  slip  of  gear  cells)  must  form  a 
closed  circuit  which  cuts  across  every  possible  chain  of  geared  cells 
extending  from  p  to  pf . 

When  a  current  flows  along  a  path  which  does  not  form  a  closed 
circuit,  then  an  increasing  ether  distortion  (electric  field)  is  pro- 
duced around  the  end  portions  of  the  path  as  explained  in  Art. 
142. 

140.  Flow  of  energy  in  the  neighborhood  of  a  wire  carrying  an 
electric  current,  (a)  Simplest  case,  ivken  no  electric  charge  resides 
on  the  surface  of  the  wire.  —  Let  AB,  Fig.  1 79,  be  a  portion  of  a 
long  wire  through  which  an  electric  current  is  flowing.  If  there 


S 

8 

8 

S 

S    ELEC.  F!EU) 

w 

RE 

CURREN- 

f                  ^ 

ENERGY  STREAM 


Fig.  1 79 


248        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

is  no  electric  charge  on  the  surface  of  the  wrire,  then  the  electric 
field  in  the  neighborhood  of  the  wire  is  parallel  to  the  wire.  The 
lines  of  force  of  the  magnetic  field,  on  the  other  hand,  encircle  the 
wire,  and  therefore  the  energy  streams  in  towards  the  wire  and  on 
all  sides,  and  is  converted  into  heat  in  the  wire. 

Let  R  be  the  resistance  of  the  wire  in  abohms  per  centimeter 
of  length,  and  let  /  be  the  current  in  the  wire,  in  abamperes, 
then  RI  is  the  intensity  of  the  surrounding  electric  field  *  in 
abvolts  per  centimeter.  According  to  Art.  55,  the  intensity  of 
the  magnetic  field  at  a  distance  of  r  centimeters  from  the  wire 
is  2ljr  gausses.  The  intensity  of  the  energy  stream  (units  of 
energy  per  unit  of  area  per  second)  at  a  distance  of  r  centi- 
meters from  the  wire  is  proportional  to  the  product  of  the  elec- 
tric field  and  magnetic  field  intensities,  and  it  may  therefore  be 
written  kx  RI  X  2f/r,  where  k  is  an  unknown  proportionality 
factor.  Multiplying  this  expression  for  the  intensity  of  the 
energy  stream  by  the  area  of  a  cylindrical  surface  /  centimeters 
in  length  and  r  centimeters  in  radius  (co-axial  with  the  wire), 
we  have  the  total  energy  per  second  streaming  in  to  /  centi- 
meters of  the  wire,  and  this  must  be  equal  to  /  x  RP.  Therefore, 
we  have 

2/ 

2Trrl  x  k  x  RI  x  — =  /  x  RI 

r 

whence 

~4?r 
Therefore  we  have 

S  =  ^.Hf  (77) 

in  which  6"  is  the  energy  in  ergs  per  second  which  streams 
across  one  square  centimeter  of  area  at  right  angles  to  a  magnetic 
field  of  which  the  intensity  is  H  gausses  and  at  right  angles  to 
an  electric  field  of  which  the  intensity  is  /  abvolts  per  centi- 
meter, H  and  /  being  at  right  angles  to  each  other. 

*  The  intensity  of  that  component  of  the  electric  field  which  is  parallel  to  the  wire. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     249 

(b)  General  casey  when  electric  charge  resides  on  the  surface  of 
the  ^vire.  —  The  component  of  the  electric  field  which  is  parallel 
to  the  surface  of  a  wire  is  always  equal  to  the  RI  drop  per 
centimeter  along  the  wire,  but  the  component  of  the  electric 
field  at  right  angles  to  the  surface  of  the  wire  may  have  any 
value  whatever,  and  the  electric  lines  of  force  which  terminate  on 
the  surface  of  the  wire  on  account  of  the  existence  of  this  normal 
component  of  electric  field  involve  a  stationary  *  electric  charge 
on  the  surface  of  the  wire.  An  example  of  this  general  case  is 
shown  in  Fig.  180.  An  electric  generator  G  delivers  current 

A 


m 

^ 

- 

— 

-> 

_ 

— 

— 

— 

-> 

— 

- 

- 

-^ 

- 

— 

—  ^ 

-^ 

t          ^ 

L 

hO* 

M'~ 

^•<\ 

cf— 

•*< 

£A~ 

/^. 

-     -H 

y? 

V^;j 

L            / 

7 

> 

] 

— 

— 

- 

- 

>- 
•> 

— 

- 

- 

- 

-• 

>~ 

s> 

F 

L 

ig. 

' 

18 

0. 

over  two  line  wires  f  A  and  B  to  a  distant  lamp  Z.  The 
electromotive  force  across  from  A  to  B  involves  the  existence 
of  an  electric  field  the  lines  of  force  of  which  trend  somewhat  as 
shown  by  the  full-line  curves  in  the  figure.  The  magnetic  field 
between  A  and  B  is  everywhere  perpendicular  to  the  plane  of 
the  paper  and  everywhere  of  the  same  intensity,  so  that  the 
energy  stream  lines  are  a  series  of  lines  which  are  everywhere  at 
right  angles  to  the  electric  lines  of  force.  The  electric  lines  of 
force  where  they  touch  A  and  B  are  slightly  inclined  to  the 

*The  electric  current  may  be  considered  to  be  a  transfer  of  electric  charge 
along  the  wire  but  the  charge  here  referred  to  has  nothing  directly  to  do  with  the 
current.  When  a  voltaic  cell  is  on  open  circuit,  the  electric  field  in  the  surrounding 
region  may  be  such  that  the  volts  per  centimeter  along  a  given  path  may  vary  in  the 
most  irregular  way  ;  but  when  this  path  is  occupied  by  a  wire  through  wriich  the 
voltaic  cell  produces  a  current,  then  the  electric  field  is  modified  by  the  charge  on 
the  surface  of  the  wire  so  as  to  make  the  component  of  the  electric  field  parallel  to 
the  wire  everywhere  equal  to  the  RI  drop  per  centimeter  along  the  wire. 

fin  order  that  Fig.  1 80  maybe  a  complete  representation,  A  and  B  must  be 
supposed  to  be  broad  metal  bands. 


250        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


surfaces  of  A  and  B  as  shown  in  the  figure,  the  degree  of  in- 
clination depending  upon  the  RI  drop  along  A  and  B. 
Therefore  the  energy,  streams  out  from  the  generator  through 
the  whole  of  the  region  between  A  and  By  and,  although  the 
energy  stream  turns  in  slightly  on  each  line  wire,  the  main  por- 
tion of  the  energy  converges  on  the  distant  lamp  L,  as  shown 
by  the  dotted  lines  in  Fig.  180.  No  attempt  is  made  in  Fig. 
1 80  to  represent  the  electric  field  distribution  in  the  neighbor- 
hood of  the  generator. 

141.  The  charge  on  a  condenser  and  its  disappearance  when  the 
condenser  plates  are  connected  by  a  wire. —  Consider  a  closed 
chain  of  gear  wheels  ABy  Fig.  181.  If  the  gears  are  allowed 

to  slip  at  any  point  s, 
the  gear  f  being  held 
stationary  and  the 
gear  e  being  turned 
in  the  direction  of  the 
arrow,  then  the  chain 
of  gears  will  be  dis- 
torted as  shown  in 
Fig.  182.  Conversely, 
a  chain  of  geared 
wheels  which  by  elas- 
tic action  tend  to  stand 
in  a  smooth  row,* 
will  be  relieved  from 
such  a  zigzag  distor- 
tion as  is  shown  in 
Fig.  182  by  permit- 
ting the  gears  to  slip  at  any  point,  s  and  the  potential  energy 
stored  in  the  distorted  chain  will  be  geared  towards  s  from  both 
sides. 

*  The  chains  of  positive  and  negative  ether  cells  are  thought  of  as  standing  in  zig- 
zag rows  when  undistorted,  as  shown  by  the  horizontal  rows  in  Fig.  173.  Here- 
after the  chains  of  ether  cells  are  to  be  thought  of  as  straight  (or  uniformly  curved) 
when  free  from  distortion,  in  order  that  the  diagrams  may  be  simpler. 


ELECTRIC    OSCILLATIONS   AND    ELECTRIC    WAVES.     251 

Let  A  and  By  Fig.  183,  be  two  metal  plates,  and  let  the 
dotted  lines  represent  closed  chains  of  geared  ether  cells,  each 
chain  being  like  Fig.  181.  Imagine  the  two  plates  A  and  B  to 
be  connected  by  a  wire,  and  an  electric  current  to  be  forced 
through  this  wire  by  means  of  a  battery,  thus  causing  the  plates 
A  and  B  to  become  charged.  The  forcing  of  the  current  through 


Fig.  183. 

the  wire  means  a  forced  slipping  of  ether  cells  at  every  point  of 
the  wire,  and  each  chain  of  geared  cells,  initially  like  Fig.  181, 
would  become  distorted  like  Fig.  182.  Throughout  the  region 
between  A  and  B  the  positive  ether  cells  would  be  displaced 
downwards  and  the  negative  ether  cells  would  be  displaced  up- 
wards, that  is,  the  region  between  A  and  B  would  become  an 
electric  field,  the  direction  of  which  would  be  outwards  from  the 
positively  charged  plate  A  and  inwards  towards  the  negatively 
charged  plate  B. 

Imagine  the  two  metal  plates  A  and  B,  Fig.  183,  to  be 
charged,  that  is,  imagine  the  chains  of  geared  ether  cells  which 
are  represented  by  the  dotted  lines  in  Fig.  183  to  be  distorted 
like  Fig.  1 8  2.  Then  a  wire  *  connected  from  A  to  B  will  cut 

*  Strictly  this  wire  should  be  thought  of  as  a  broad  sheet  of  metal  of  which  the 
sectional  view  is  shown  in  Fig.  183.  See  footnote  on  page  246. 


252        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

across  every  one  of  the  distorted  chains  of  geared  ether  cells, 
slipping  will  begin  at  every  point  on  the  wire,  each  distorted  chain 
of  cells  will  begin  to  be  relieved  from  distortion,  the  energy  of 
each  distorted  chain  will  be  transmitted  along  the  chain  to  the 
wire  where  it  will  appear  as  heat,  and  the  entire  region  between 
and  surrounding  the  metal  plates  A  and  B.  will  be  relieved 
from  the  electrical  stress.  The  explanation  here  given  of  the  en- 
tire relief  of  the  electrical  stress  between  two  plates  by  the  estab- 
lishment of  a  conducting  line  (line  of  slip)  between  them,  applies 
to  two  adjacent  oppositely  charged  bodies  of  any  shape.  An 
electric  spark  is  a  line  of  slip  (a  conducting  line)  and  the  energy 
of  the  electic  field  flows  in  upon  the  spark  as  it  does  upon  a  wire. 
The  slipping  of  the  ether  cells  in  a  conductor  is  imagined  to  be 
opposed  by  a  fractional  drag  very  much  as  if  the  gear  teeth  of  the 
ether  cells  in  a  metal  were  made  of  a  viscous  substance  like  pitch. 

142,  The  electric  oscillator.  —  Let  A  and  B,  Fig.  184,  be  two 
metal  balls  connected  to  two  short  rods  between  which  is  an  air 
gap.  Imagine  charge  to  have  been  collecting  on  A  and  B 


Fig.  184. 


(positive  on    A,    negative  on    B]    until  a  spark  jumps  across  the 
air  gap,  thus  establishing  a  conducting  path  from    A    to    B   and 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     253 

causing  A  and  B  to  discharge.  This  discharge,  however,  is 
usually  oscillatory  like  the  movements  of  a  spring  which  is 
pulled  to  one  side  and  suddenly  released,  as  follows  :  Consider  a 
chain  of  geared  ether  cells  which  when  undistorted  lies  along  the 
dotted  line  in  Fig.  1 84,  this  dotted  line  being  everywhere  per- 
pendicular to  the  lines  of  force  of  the  electric  field.  When  A 
is  positively  charged  this  chain  is  distorted  as  shown  (in  part),  but, 
inasmuch  as  it  is  a  closed  chain,  its  distortion  is  fixed,  as  ex- 
plained in  connection  with  Fig.  182.  When  a  spark  is  formed 
across  the  air  gap,  however,  a  line  of  slip  is  established  across  the 
distorted  chain,  and  the  distortion  disappears  as  explained  in  Art. 
141.  What  is  said  of  the  single  chain  of  ether  cells  is  true  of 
every  chain  which  surrounds  A  or  B. 

If  the  slip  which  relieves  the  distortion  of  the  chain  of  ether 
cells  takes  place  with  great  friction  (high  electrical  resistance  of 
conducting  path  formed  by  the  spark),  the  cells  near  the  spark 
begin  turning  slowly  and  the  entire  energy  of  the  distorted  chain 
is  geared  into  the  gap  and  converted  at  once  into  heat.  If  the 
slip  which  relieves  the  distortion  of  the  chain  of  ether  cells  is 
almost  frictionless  (low  electrical  resistance  of  the  conducting  path 
formed  by  the  spark),  then  the  energy  of  the  distorted  chain  is 
used  mostly  in  overcoming  the  inertia  of  the  cells  as  they  are  set 
rotating,  and  after  a  very  short  interval  of  time  the  whole  of  the 
electrical  energy  will  have  been  converted  into  kinetic  energy  of  the 
rotating  cells  (magnetic  energy).  During  this  conversion  the 
energy,  streaming  along  the  dotted  lines  in  Fig.  1 84,  largely  dis- 
appears from  the  regions  ee  and  ee,  and  is  distributed  mainly  in 
the  region  mm.  When  the  chain  of  ether  cells  has  been  relieved 
from  distortion,  the  rotatory  motion  of  the  ether  cells  in  the  region 
mm  will  have  reached  a  maximum,  and  the  cells  will  continue  to 
rotate  because  of  their  momenta,  thus  producing  a  reversed  distor- 
tion of  each  chain  of  ether  cells.  At  the  same  time  the  energy 
will  stream  back  from  the  region  mm  to  the  regions  ee  and  ee, 
being  converted  again  into  potential  energy  of  ether  distortion,  and 
the  balls  A  and  B  will  be  charged  in  a  reversed  sense.  This  re- 


254        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

versed  distortion  of  the  chains  of  ether  cells  is  then  relieved  by  a 
reversed  slip  (a  reversed  current  in  the  rods  and  along  spark), 
and  the  above  described  action  is  repeated  over  and  over  again 
until  the  original  energy  of  the  electrical  field  has  been  dissipated. 
The  oscillatory  changes  above  described  take  place  so  rapidly 
that  the  portions  of  the  distorted  ether  which  are  remote  from  the 
oscillator  AB,  Fig.  184,  do  not  follow  the  changes  promptly. 
This  gives  rise  to  electrical  waves  the  nature  of  which  at  a  dis- 
tance from  the  oscillator  is  explained  in  a  subsequent  article. 

143.  Examples  of  electric  oscillators.  —  The  type  of  electric 
oscillator  which  is  described  in  Art.  142  was  devised  by  Hertz 
and  used  by  him  in  his  celebrated  experimental  researches  on 
electric  waves  in  1887.*  An  electric  oscillator  essentially  similar 
to  the  Hertz  oscillator  is  employed  as  the  sending  device  in 
electric-wave  telegraphy,  wireless  telegraphy  so-called,  as  de- 
scribed in  Appendix  D. 

Almost  every  electric  spark  discharge  is  oscillatory  in  character 
as  may  be  shown  by  photographing  the  spark  upon  a  rapidly 
moving  photographic  plate.  Thus,  a  sharp  flash  of  lightning 
when  photographed  by  means  of  a  rapidly  swinging  camera 
generally  shows  several  parallel  flashes  very  close  together  on 
the  photographic  plate.  The  number  of  oscillations  per  second 
of  an  electric  discharge  is,  however,  generally  so  great  that  the 
sound  of  the  spark  cannot  be  distinguished  from  a  sharp  snap  or 
click.  According  to  the  principles  enunciated  in  Appendix  E, 
however,  it  is  evident  that  the  number  of  oscillations  per  second 
can  be  reduced  to  any  desired  value  by  increasing  the  inductance 
of  the  circuit  through  which  the  discharge  takes  place  and  by 
increasing  the  capacity  of  the  condenser  in  which  the  charge  is 
stored.  Thus,  Fig.  185  shows  a  battery  of  Leyden  jars  JJ 
arranged  to  discharge  across  an  air  gap  g  and  through  a  coil  of 
wire  L.  The  sound  produced  by  the  spark  in  this  case  is  a  high 
pitch  musical  tone  of  very  short  duration  like  the  ringing  sound 

*  These  researches  are  described  in  Hertz's  book  on  Electric  Waves,  English 
translation  published  by  The  Macmillan  Company. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     255 


which  is  produced  by  striking  an  anvil  with  a  hammer.  The 
pitch  of  the  "ringing  spark"  may  be  raised  by  decreasing  the 
number  of  turns  of  wire  in  the  coil  Z,  or  by  decreasing  the 


fo  electric 
machine 


Fig.  185. 

number  of  Leyden  jars  in  the  battery  JJ.  The  oscillatory  char- 
acter of  the  spark  across  the  gap  g  may  be  shown  by  viewing 
it  in  a  rotating  mirror,  and  in  this  way  ten  or  more  images  of  the 
spark  may  be  seen  side  by  side  at  each  discharge  of  the  battery 
of  Leyden  jars. 

The  oscillatory  discharge  of  a  condenser  through  a  coil  of  wire 
is  utilized  in  a  type  of  induction  coil  which  is  due  to  Nikola 
Tesla.  A  helix  PP,  Fig.  186,  of  ten  or  fifteen  turns  of  coarse 

wire  is  connected  to  the 

S 
Q. 


TO  TRANSFORMER 


TO  TRANSFORMER 


Fig.  186. 


terminals  CD  of  a 
charged  condenser  AB, 
or  battery  of  Leyden 
jars,  with  a  spark  gap  in 
the  circuit  at  g.  The 
condenser  is  connected 
to  the  secondary  of  a  high  voltage  transformer,  each  high  volt- 
age impulse  of  the  transformer  charges  the  condenser  until  the 
air  gap  at  g  breaks  down,  and  then  the  condenser  charge 
surges  back  and  forth  through  the  helix  PP  until  the  energy  of 
the  charge  is  dissipated.  A  jet  of  air  *  issues  from  a  nozzle  J 
and  blows  away  the  air  which  has  been  heated  and  ionized  by 

*  By  using  zinc  terminals  for  the  spark  g   and  by  placing  two  or  three  very  short 
spark  gaps  in  series  the  air  jet  becomes  unnecessary. 


2  $6        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  spark.  Then  charge  can  again  accumulate  on  the  condenser 
until  a  new  discharge  takes  place.  The  successive  discharges 
may  be  as  frequent  as  .several  thousand  per  second  (a  number  of 
successive  discharges  taking  place  during  each  high  voltage  im- 
pulse of  the  charging  transformer),  and  the  oscillations  of  each 
discharge  may  be  at  the  rate  of  a  million  or  more  per  second. 

A  second  helix  55  of  several  hundred  turns  of  wire  sur- 
rounds the  helix  PP  (not  so  shown  in  the  figure),  that  is,  the  coils 
PP  and  55  constitute  the  primary  and  secondary  coils  of  an 
induction  coil.  The  rapidly  oscillating  current  in  PP  due  to 
the  discharge  of  the  condenser  induces  very  large  electromotive 
forces  in  55  and  produces  long  sparks  between  the  terminals 
of  55. 

A  very  striking  property  of  the  discharge  from  55,  which  is 
due  to  its  high  frequency,  is  that  it  traverses  only  the  surface 
layers  of  a  conductor  and  it  may  therefore  be  passed  through 
(over)  the  human  body  with  impunity. 

Leyden  jars  as  oscillators  and  resonators.  —  Similar  circuits  may 
be  connected  to  two  Leyden  jars  so  that  the  oscillations  which 
occur  when  one  Leyden  jar  discharges  through  its  circuit  are  in 
unison  with  the  proper  oscillations  of  the  closed  circuit  of  the  other 
jar,  so  that  the  inducing  action  on  the  circuit  of  the  second  jar  is 
cumulative.  An  instructive  experiment  is  the  following  :  A  Ley- 
den jar  is  connected  to  a  vertical  rectangular  circuit  of  wire  ww  as 
shown  in  Fig.  187,  and  an  electric  machine  repeatedly  charges  the 
jar  until  it  discharges  across  the  air  gap  g  and  through  the  circuit 
ww.  This  discharge  is  oscillatory  in  character  and  it  has  a 
definite  frequency.  A  second  jar  similar  to  the  first  is  short-cir- 
cuited by  a  vertical  rectangular  wire  frame  ww  as  shown  in  Fig. 
1 88,  and  placed  along  side  of  the  arrangement  shown  in  Fig. 
187.  By  adjusting  the  size  of  the  circuit  in  Fig.  1 88,  the  free 
period  of  oscillation  of  this  circuit  may  be  made  to  coincide  with 
the  period  of  oscillation  of  the  circuit  in  Fig.  187,  and,  when 
this  condition  is  reached,  the  induced  oscillations  in  the  circuit 
become  sufficiently  intense  to  produce  a  spark  across  the  air  gap 


ELECTRIC   OSCILLATIONS  AND    ELECTRIC   WAVES.     257 


a.  This  experiment  illustrates  the  phenomenon  of  electric  reso- 
nance. Each  oscillation  of  the  circuit  in  Fig.  187  induces  a 
slight  electromotive  force  in  the  circuit  of  Fig.  1  88,  these  succes- 


to  electric 


machine 


to  electric 


machine 


wire 


rod 


w    wire 


Fig.  187. 


rod 


wire 


w 


w      wre 


Fig.  188. 


rod 


sive  electromotive  forces  are  in  unison  with  the  proper  period  of 
oscillation  of  the  circuit  in  Fig.  188,  and  therefore  their  effect  is 
cumulative. 

The  Tesla  induction  coil  is  usually  arranged  so  that  the  induc- 
tance of  its  primary  circuit  can  be  adjusted,  thus  altering  the 
frequency  of  the  oscillatory  discharges  through  the  primary  coil. 
Then  by  adjusting  the  inductance  of  the  primary  until  the  fre- 
quency of  oscillation  of  the  primary  circuit  is  the  same  as  the 
frequency  of  oscillation  of  the  secondary  circuit,  the  successive 
surges  of  current  in  the  primary  coil  become  cumulative  in  their 
effect  on  the  secondary  (resonance). 

144.  Water  waves  in  a  canal.  —  Before  attempting  to  describe 
electrical  waves,  it  is  desirable  to  consider  some  of  the  phe- 
nomena presented  by  water  waves.  A  water  wave  consists  of  a 
moving  hill  of  water,  a  given  particle  of  water  is  set  in  motion 
when  the  wave  reaches  it,  and  comes  immediately  to  rest  after 
the  wave  has  passed.  What  supports  the  hill  of  water,  and 
what  produces  the  unbalanced  force  which  causes  the  water  to 
18 


258         ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


gain  velocity  and  lose  it  again  during  the  passage  of  the  wave  ? 
A  wave  always  consists  of  tivo  elements  whicli  travel  along  together ', 
a  local  distortion  of  tlie  medium  and  a  local  state  of  motion  of  the 
medium,  the  forces  which  are  associated  with  the  distortion  are  the 
forces  which  produce  the  motion  ;  this  production  of  motion  involves 
acceleration  and  the  reaction  of  the  acceleration  gives  rise  to  the 
forces  which  produce  distortion.  The  distortion  creates  the  mo- 
tion and  the  motion  creates  the  distortion  as  they  both  travel 
along  together.  The  two  are  mutually  dependent. 

A  consideration  of  the  simplest  kind  of  water  waves  in  a  canal, 
namely,  the  kind  in  which  the  only  perceptible  motion  of  the 
water  in  the  wave  is  a  uniform  horizontal  floiu,  will  serve  better 
than  anything  else  as  an  introduction  to  the  discussion  of  electric 
waves.  Consider  a  canal  of  rectangular  section  which  is  filled  to 
a  depth  x  with  still  water.  Imagine  a  gate  to  be  moved  slowly 
along  the  canal  at  velocity  vt  as  shown  in  Fig.  1 89.  The  water 
next  the  gate  is  set  in  motion,  and  in  being  set  in  motion  it  heaps 
up  to  a  definite  depth  x  +  h  ;  and  a  wave  of  starting  W  moves 

along  the  canal  at  a  definite 
velocity  V.  If  the  gate  is  sud- 
denly  stopped,  the  wave  of 
starting  W  continues  to  move 
as  before,  the  water  next  to  the 
gate,  in  being  stopped,  drops 
to  its  normal  depth  xy  and  a 
wave  of  arrest  W9  moves  along  the  canal  as  shown  in  Fig. 
190.  The  elevation  h  of  the  water  in  the  wave  is  supposed  to 
be  small. 

The  uniformly  moving  and  uniformly  elevated  body  of  water 
A,  Fig.  190,  constitutes  what  is  called  a  complete  wave,  or  simply 
a  wave.  The  water  in  front  of  the  wave  is  continually  set  in 
motion  at  velocity  v  and  raised  to  the  depth  x  +  h.  The  water 
in  the  back  part  of  the  wave  is  continually  brought  to  rest  and 
lowered  to  the  normal  depth  x  of  the  water  in  the  canal. 
Thus,  the  state  of  motion  which  constitutes  the  wave  A  travels 


still  water 


Fig.   189. 


ELECTRIC   OSCILLATIONS   AND   ELECTRIC   WAVES.     259 


along   the   canal   without   changing  its  character,  friction  being 
neglected. 

An  essential  feature  of  any  wave  which  moves  along  without 
changing  its  shape  is  that  the  kinetic  energy  is  equal  to  the  potential 
energy  in  the  wave  at  each  point.     Thus,  the  kinetic  energy  of  ' 
the  water  wave   A,    Fig.    1 90,  due  to  the  uniform  velocity    v   of 


still  water 


moving 
W      water 


'.- ':  ':'.••'•  v '-.  :;;.'.*  Jf  :*•  :-v.-i  IfcC-^^^y 


still  water 


stationary 


moving  'water 


W   still  water      § 


gate 


Fig.   190. 

the  water  in  the  wave  is  equal  to  the  potential  energy  due  to  the 
elevation   7z.*     When  the  potential  energy  in  a  wave  is  equal  to 

*  The  following  derivation  of  the  velocity  of  a  water  wave  in  a  canal  shows  the 
significance  of  equality  of  potential  and  kinetic  energy.  This  discussion  is  based 
upon  a  slight  modification  of  the  conditions  shown  in  Fig.  189,  as  follows  :  Water  of 
depth  x  flows  along  a  canal  of  rectangular  section  at  a  uniform  velocity  (small)  of 
v  centimeters  per  second.  A  gate  is  suddenly  closed  as  shown  in  Fig.  191  ;  the 
moving  water,  in  being  brought  to  rest  against  the  gate,  heaps  up  to  a  depth  x  -j-  h  ; 

and  a  wave  of  arrest  W,  Fig.  191, 
moves  along  the  canal  at  a  definite  ve- 
locity V.  The  action  involved  in  Fig. 
191  is  identical  to  the  action  involved 
in  Fig.  189.  In  fact,  Fig.  189  can  be 
converted  into  Fig.  191,  by  imagining 
everything  in  Fig.  189  to  be  moving  to 
the  right  at  velocity  v.  The  discussion 
of  Fig.  191  is  simpler  than  the  discus- 
sion of  Fig.  189  because  the  potential  energy  is  in  one  portion  of  the  water  and 
the  kinetic  energy  is  in  another  portion,  whereas  in  Fig.  189  the  potential  energy  and 
the  kinetic  energy  are  both  in  one  portion  of  the  water.  Let  b  be  the  breadth  of  the 
canal.  Consider  a  transverse  slice  of  water  one  centimeter  thick.  The  volume  of 
this  slice  is  bx  cubic  centimeters  and  its  mass  is  dbx  grams,  where  d  is  the  density 
of  the  water  in  grams  per  cubic  centimeter.  Therefore  the  kinetic  energy  of  thisslice 
of  water  when  it  is  moving  at  a  velocity  of  v  centimeters  per  second  is  \dbxv2. 
When  the  wave  of  arrest  Wt  Fig.  191,  reaches  the  slice  of  water  under  consideration, 
the  slice,  as  it  comes  to  rest,  is  squeezed  together  and  increased  in  depth  to  x  -j-  h. 
The  slice  is  decreased  in  thickness  in  proportion  to  its  increase  in  depth,  so  that  its 


Fig.  191 


260        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  kinetic  energy,  we  have  what  is  called  a  pure  wave,  and  when 
the  potential  energy  in  a  wave  is  not  equal  to  the  kinetic  energy 
the  wave  is  called  an  impure  wave. 

The  behavior  of  an  impure  wave  pulse  in  a  canal  may  be  stated 
by  considering  an  extreme  case  of  an  impure  wave  as  follows  : 
Consider  an  elevated  portion  of  still  water  in  a  canal  as  shown 


still  water 


still  water  {.  ......  ;.  ...... ;  ..j    still  wa 


still  water 


Fig.  192. 


Fig.  193. 


thickness  is  reduced  to  x/(jc-\-&)  or  to  (i — hjx]  of  a  centimeter,  h  being  very 
small.  Therefore,  the  decrease  of  thickness  is  h\x  of  a  centimeter.  The  force 
acting  to  reduce  the  thickness  of  the  slice  is  to  be  considered  as  that  force  which  is 
due  to  the  increase  of  pressure  in  the  water  produced  by  the  increasing  depth  h. 
This  increase  of  pressure  is  equal  to  hdg  dynes  per  square  centimeter  when  the  slice 
has  reached  its  greatest  depth,  so  that  the  average  increase  of  pressure  due  to  increas- 
ing depth  is  \hdgy  which  produces  over  the  face  of  the  slice  a  force  equal  to 
\hdg  X  bxy  and  the  product  of  this  force  and  the  decrease  of  thickness  of  the  slice 
gives  the  work  done  in  decreasing  its  thickness.  This  work  must  be  equal  to  the 
original  kinetic  energy  of  the  slice,  so  that 


Consider  the  instant  /  seconds  after  the  closing  of  the  gate  in  Fig.  191.  The  wave 
of  arrest  W  has  reached  the  distance  Vt  from  the  gate,  and  the  excess  of  water  that 
is  represented  by  the  raising  of  the  water  level  (=  Vfy^h  X  ^  cubic  centimeters)  is 
the  amount  of  water  supplied  by  the  flow  of  the  canal  in  /  seconds  (=^bxvt  cubic 

centimeters).     Therefore 

Vthb  —  bxvt 
or 


Substituting  the  value   v   from  equation  (i)  in  equation  (ii),  we  have 


(H) 


Therefore  the  velocity  of  progression  of  a  wave  in  a  canal  is  equal  to  the  velocity 
gained  by  a  body  in  falling  freely  through  the  distance  xjz. 


ELECTRIC   OSCILLATIONS   AND   ELECTRIC   WAVES.     261 

in  Fig.  192.  This  body  of  elevated  water  is  an  impure  wave  in- 
asmuch as  its  velocity  of  flow  v  is  zero,  and  therefore  its  potential 
energy  of  elevation  cannot  be  equal  to  its  kinetic  energy  of  flow. 
Such  an  elevated  portion  of  still  water  breaks  up  into  two  oppo- 
sitely moving  pure  waves,  and  the  initial  stage  of  this  process  of 
breaking  up  is  indicated  in  Fig.  193. 

When  a  wave  like  A,  Fig.  190,  travels  along  a  canal,  the 
velocity  of  flow  v  is  continually  decreased  by  friction,  whereas 
there  is  no  action  tending  to  reduce  the  elevation  h.  Therefore 
that  portion  of  the  elevation  which  is  in  excess  of  what  is  required 
to  give  a  pure  wave  with  what  remains  of  the  velocity  of  flow, 
behaves  exactly  like  the  elevation  A  in  Fig.  192,  that  is,  this 
excess  of  elevation  breaks  up  into  two  pure  waves  a  and  b,  Fig. 
193,  the  portion  a  merges  with  the  original  wave  A  and  the 
portion  b  shoots  backwards. 

The  upper  part  of  Fig.  194  represents,  on  an  exaggerated 
scale,  the  elevated  portion  of  water  in  a  pure  wave.  The  velocity 


-y- 

h 

A 

^  direction  of  progression        j 

^ 

head 


tail  of  wave 

Fig.  194. 


of  flow  v  in  this  wave  is  continually  reduced  by  friction  as  the 
wave  travels  along  the  canal,  the  excess  of  elevation  which  is 
being  thus  continually  left  in  the  wave  causes  a  long  drawn-out 
wave  to  shoot  backwards,  and  after  a  time  the  wave  has  the 
form  shown  in  the  lower  portion  of  Fig.  194,  The  head  of  the 
wave  is  greatly  reduced  in  intensity  (energy  value)  partly  be- 
cause of  the  loss  of  energy  by  friction  and  partly  because  of  the 


262        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

carrying  of  energy  backwards  into  the  tail  of  the  wave.  After 
a  given  interval  of  time  the  tail  of  the  wave  has  a  total  length 
2  Vt  where  V  is  the  -velocity  gf  progression  of  the  wave. 

If  a  canal  is  filled  brimful  of  water  so  that  the  elevation  of  the 
water  level  causes  an  overflow,  or  spill,  the  tendency  is  for  a 
wave  to  remain  pure,  and  therefore  to  be  ^propagated  without 
change  of  shape,  because  the  elevation  is  reduced  by  spill  and 
the  velocity  of  flow  v  is  reduced  by  friction.  This  is  precisely 
analogous  to  the  action  which  takes  place  on  a  poorly  insulated 
telephone  line  and  which  causes  such  a  telephone  limit  to  transmit 
speech  more  distinctly  than  if  it  were  thoroughly  insulated. 

145.  The  electromagnetic  wave.  —  An  electromagnetic  wave 
consists  of  a  state  of  ether  distortion  and  a  state  of  ether  motion 
traveling  along  together  and  mutually  sustaining  each  other. 
The  ether  distortion  is  electric  field  and  the  ether  motion  is  mag- 
netic field.  A  layer  of  electric  field  unsustained  breaks  up  into 
two  electromagnetic  waves  just  as  the  elevated  portion  of  water 
in  Fig.  192  breaks  up  into  two  water  waves. 

The  action  which  takes  place  in  an  electromagnetic  wave  may 
be  clearly  understood  with  the  help  of  Maxwell's  conception  of 
the  electromagnetic  field.  It  is  desirable  to  consider  the  case  of 
an  electric  wave  which  moves  along  between  two  wires  (or  broad 
sheets  of  metal)  which  bound  the  electric  wave  very  much  as  a 
speaking  tube  bounds  a  sound  wave  which  passes  through  it. 

Figure  195  shows  two  broad  sheets  of  metal  with  an  electro- 
magnetic wave  pulse  traveling  along  between  them  at  velocity 
K  The  fine  vertical  lines  represent  the  electric  field  which  is 
towards  the  top  of  the  page,  and  the  dots  represent  the  lines 
of  force  of  the  magnetic  field  which  is  perpendicular  to  the  plane 
of  the  paper  and  directed  towards  the  reader.  A  single  chain 
of  geared  cells  is  shown  in  the  figure,  although  a  complete  repre- 
sentation of  what  takes  place  in  the  wave  would  necessitate  the 
showing  of  great  numbers  of  horizontal  chains  of  geared  ether 
cells  every  one  of  which  would  be  exactly  similar  to  the  one 
shown  in  Fig.  195.  Within  the  region  of  the  wave  the  ether 


ELECTRIC   OSCILLATIONS   AND   ELECTRIC   WAVES.    263 

cells  are  all  in  uniform  rotation  as  indicated  by  the  small  curved 
arrows,  and  within  the  region  of  the  wave  the  chains  of  cells  are 
all  distorted,  positive  cells  being  displaced  upwards  with  respect 
to  the  negative  cells,  as  shown  in  Fig.  195. 

electric  current 


> 

; 

~~"*negative  charge 

v      **- 

T 

$H 

*= 

=5. 

£= 

=5. 

>SZ 

^      ?' 

wire 

,    •• 

• 
• 

o 

•    ( 

• 
» 

.   • 
• 
• 

.    •  v 
• 

» 

>    •    . 
• 
• 

w  •   '' 
• 
4 

+  ' 

V       s 

"^ 
,__  ^positive  charge' 

-j- 

f 

+ 

-f- 

-|- 

electric  current           *~               wirv 

Fig.  195. 

Throughout  the  middle  portion  of  the  wave  each  rotating  cell 
is  acted  upon  by  equal  and  opposite  torques  by  the  adjacent  cells 
ahead  of  it  and  behind  it,  as  explained  in  connection  with  Fig. 
175.*  Therefore  all  the  cells  in  the  middle  portion  of  the  wave 
continue  to  rotate  at  unchanging  speed,  and  the  zigzag  distortion 
of  the  chain  of  cells  remains  unchanged  in  the  middle  portion  of 
the  wave  The  cell  d,  however,  exerts  an  unbalanced  torque 
upon  the  cell  ft  as  indicated  by  the  dotted  arrow  Tf,  and  this 
torque  quickly  sets  the  cell  f  into  rotation.  Also  the  cell  b 
exerts  an  unbalanced  torque  T  upon  the  cell  c  which  quickly 
stops  the  rotation  of  the  cell  c.  Thus  the  combined  state  of 
motion  and  distortion  of  the  ether  cells  between  c  and  f  travels 
to  the  right. 

The  terminating  of  the  electric  lines  of  force  on  the  wires  (or 
metal  sheets)  which  bound  the  electric  wave  constitutes  electric 

*  Figure  195  represents  what  maybe  called  a  rectangular  electromagnetic  wave 
pulse  throughout  which  the  electric  field  is  uniform  and  throughout  which  the  mag- 
netic field  is  uniform. 


264        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

charges,  positive  on  the  lower  wire  and  negative  on  the  upper 
wire,  in  Fig.  195.  It  is  evident,  furthermore,  that  the  uniform 
rotation  of  the  ether. cells  in  the  region  of  the  wave  involves  the 
slipping  of  the  ether  cells  where  they  come  in  contact  with  the 
sheets  of  metal  which  bound  the  wave.  This  slipping  constitutes 
an  electric  current  which  flows  to  the  left  in  the  upper  wire  and 
to  the  right  in  the  lower  wire  in  Fig.  195. 

Figure  195  represents  what  is  called  a  rectangular  wave  pulse. 
Fig.  196  shows  what  takes  place  when  a  simple  train  of  electro- 
magnetic waves  travel  along  between  two  broad  metal  sheets. 

H  inward  1 


•«- 
wire 

•<• 

-  -» 

—  > 

4- 

->- 

h  + 

—  m~ 

4-+J- 

•<- 

<--< 

—  •«- 

-.? 

\— 

+ 

-^ 

h- 

> 
t. 

I 

,  ' 

•  '•*! 

•  •'. 

v 

. 

-'•  : 

> 

. 

• 

•1 

I 

• 

i 

. 

: 

''  .  .  ',:• 

'.•      ' 

• 

• 

• 

!•    . 

• 

• 

. 

' 

• 

• 

wire 


;:;+++: 

direction  of  progression 


wire 


axis 


Fig.  196. 

Hertzes  experiments  with  electric  waves.*  —  The  oscillator  used 
by  Hertz  consisted  of  two  brass  rods  A  and  B  with  an  air  gap 
g,  as  shown  in  Fig.  197.  These  two  rods  were  connected  to 
the  terminals  of  an  induction  coil  as  indicated,  at  each  impulse 
of  electromotive  force  from  the  induction  coil  a  spark  breaks 

*  These  experiments  were  described  originally  in  Wiedemanri* s  Annalen.  A  very 
complete  description  of  them  may  be  found  in  Hertz's  book  on  Electric  Waves,  Eng- 
lish translation  published  by  The  Macmillan  Company. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     265 


across  the  gap  g  and  the  discharge  surges  back  and  forth  along 
the  rods  until  the  energy  of  the  charge  is  dissipated. 

The  resonator.  —  The  elec- 
tric waves  were  detected  in 
Hertz's  original  experiments 
by  means  of  an  arrangement 
similar  to  the  oscillator,  but 
with  a  shorter  spark  gap  and  _ 
without  connections  to  an  in-  ^ 

duction  coil.  This  arrange- 
ment, which  is  called  the  resonator,  has  the  same  period  of  oscil- 
lation as  the  oscillator  so  that  the  action  upon  it  of  the  train  of 
waves  from  the  oscillator  is  cumulative,  causing  it  to  oscillate  in 
sympathy  with  the  oscillator  just  as  one  tuning  fork  vibrates  in 
unison  with  a  similar  one  which  is  set  vibrating  with  a  hammer 
blow.  The  oscillations  of  the  resonator  were  indicated  by 
minute  sparks  in  its  gap  g,  Fig.  198. 

The  reflectors.  —  The  waves  which  emanate  from  the  Hertz 
oscillator  are  very  weak  at  any  considerable  distance,  and  their 
action  upon  the  resonator  may  be  greatly  intensified  by  the  use 
of  parabolic  reflectors.  The  oscillator  and  the  resonator  were 


oscillator 


Oscillator. 
to  coil 


to  coil 


1 

j 

I 
1 

i 

I 

I 
1 

I  I 

1 

1 

ca 

>es 

• 

1 
1 
1 
1 
I 

1 

1 
t 

1 
1 
1 

op 

vit 

f 

>w 

1 

1 

1 

1 
1 

w 

av 

1 

es\ 

\ 

[•^ 

I! 

sic 

Fi 

ei 

g. 

? 
\ 

new 

98. 

resonator 


^resonator 


266        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


placed  along  the  respective  focal  lines  of  two  parabolic  cylinders 
made  of  sheet  metal,  as  shown  in  Fig.  1 98  and  the  resonator  was 
arranged  so  that  its  spark  gap  was  behind  the  mirror  and  thus 
easily  visible. 

Reflection  of  electric  waves.  —  When  the  oscillator  and  resona- 
tor are  arranged  as  shown  in  Fig.  198,  a  very  distinct  effect  of 
the  resonator  is  produced  when  the  oscilla- 
tor is  active,  the  waves  from  the  oscillator 
being  concentrated  upon  the  resonator  by 
the  action  of  the  two  parabolic  reflectors. 

When  arranged  as  shown  in  Fig.  199, 
AB  being  a  plain  sheet  of  metal,  and  the 
angles  <f>  being  equal,  a  very  distinct  effect 
on  the  resonator  is  produed. 

Refraction  of  electric  waves. — When  the 
oscillator  and  resonator  are  arranged  as 
shown  in  Fig.  200,  in  which  PP  repre- 
sents a  large  prism,  of  asphaltum  or  paraffine,  a  very  distinct 
effect  is  produced  upon  the  resonator. 

Polarization  of  electric  waves.  —  A  frame  strung  with  a  grating 
of  fine  metal  wire  acts  as  a  good  reflector  for  the  waves  from  a 


Fig.  199. 


resonator 


Fig.  200. 


Hertz  oscillator,  when  the  wires  of  the  grating  are  parallel  to  the 
axis  of  the  oscillator.  When  the  wires  of  the  grating  are  at  right 
angles  to  the  axis  of  the  oscillator,  the  waves  pass  through  the 
grating  without  perceptible  diminution  in  intensity.  Therefore 
the  waves  from  a  Hertz  oscillator  are  plane  polarized. 

Stationary  electric  waves.  —  If  the  plane  waves  from  the  oscil- 


ELECTRIC   OSCILLATIONS  AND    ELECTRIC   WAVES.     267 


lator  and  its  parabolic  mirror  are  allowed  to  fall  perpendicularly 
upon  a  plane  sheet  of  metal  AB,  as  shown  in  Fig.  201,  the 
resonator  is  not  acted  upon  if  it  is  placed  at  certain  points 
n,  n* ',  n' ' ',  and  so  on,  whereas  the  resonator  is  acted  upon  if  it  is 
placed  at  positions  intermediate  between  these  points.  The 


resonator 


'oscillator 


I  I 

,waves 


/I'' 


Fig.  201. 


B 


reflected  waves  from  AB,  Fig.  201,  form  with  the  advancing 
waves  a  stationary  wave  train  of  which  nodes  are  situated  at  the 
points  ny  n'  ,  n"  y  and  the  antinodes  at  the  points  aa. 

146.  The  law  of  induced  electromotive  force  and  its  bearing  upon 
electromagnetic  wave  motion.  —  Let  H  be  the  intensity  in  gausses 
of  the  magnetic  field  in  the  region  of  the  wave  shown  in  Fig.  195, 
let  f  be  the  intensity  of  the  electric  field  in  abvolts  per  centi- 
meter, and  let  /  be  the  distance  across  from  wire  to  wire  (sheet 
to  sheet).  The  sidewise  motion  of  the  magnetic  field  at  velocity 
V  induces  an  electromotive  force  in  the  region  of  the  wave  and 
this  electromotive  force  in  abvolts  is  given  by  the  equation 


as  explained  in  Art.  64.  Therefore  the  electric  field  intensity  in 
the  wave  (£//)  is  given  by  the  equation 

f=HV  (78) 

in  which  f  is  expressed  in  abvolts  per  centimeter,  H  is  ex- 
pressed in  gausses,  and  V  is  expressed  in  centimeters  per 
second. 


268        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Calculation  of  velocity  of  progression  of  the  electromagnetic 
wave.  —  The  intensities  of  the  mutually  dependent  electric  and 
magnetic  fields  which  -constitute^  pure  electromagnetic  wave  must 
satisfy  two  conditions,  namely,'  (a)  the  magnetic  energy  per  unit 
volume  in  the  wave  must  be  equal  to  the  electric  energy  per  unit 
volume  in  the  wave,*  and  (b)  the  velocity  of  the  wave  must  be 
such  as  to  satisfy  equation  (78),  so  that  the  electric  field  may  be 
wholly  sustained  by  the  inducing  action  of  the  moving  magnetic 
field. 

The  magnetic  energy  in  ergs  per  cubic  centimeter  in  a  wave  is 
equal  to  H^J^TT  according  to  equation  (27),  the  intensity  H  of 
the  magnetic  field  being  expressed  in  gausses.  The  electric 
energy  per  unit  volume  in  a  wave  is  given  by  equation  (75),  in 
which  equation  the  energy  is  expressed  in  joules  per  cubic  centi- 
meter and  the  electric  field  intensity  is  expressed  in  volts  per 
centimeter.  Reducing  to  c.g.s.  units  (energy  in  ergs  per  cubic 
centimeter  and  electric  field  intensity  in  abvolts  per  centimeter) 
we  have 

fzj(2Bx  io9) 

as  the  expression  for  the  electric  energy  in  ergs  per  cubic  centi- 
meter. Therefore  the  first  condition  above  mentioned  gives  the 
equation 

fP_        f2 

87T~2£    X    IO9 

Therefore  solving  equations  (78)  and  (79)  for    Vt     we  have 

F2  =  -  x  io9  (80) 

47T 

but  the  factor  B  is  equal  to  1.131  X  io13,  according  to  Arts. 
91  and  98.  Therefore  we  have 

F=  2.996  x  ioloCm-'  (Si) 

sec. 

The  velocity  of  an  electric  wave  thus   calculated  is  identically 

*  See  footnote  to  Art.  144. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.    269 

-equal  to  the  velocity  of  light  as  determined  by  direct  observation. 
Therefore  the  most  accurate  method  for  determining  the  value  of 
the  constant  B  as  used  in  Arts.  91  to  98  is  to  calculate  its  value 
from  the  observed  value  of  V  using  equation  (Si). 

The  identity  of  the  velocities  of  electromagnetic  waves  and  of 
light  waves  was  first  pointed  out  by  Maxwell  and  it  is  now  uni- 
versally conceded  that  light  waves  are  electromagnetic  waves. 

147.  Electric  wave  distortion.  —  So  long  as  the  electric  and 
magnetic  field  intensities  in  the  wave  which  is  shown  in  Fig.  195 
continue  to  satisfy  equation  (79),  the  electromagnetic  wave  remains 
pure  and  it  does  not  change  its  shape  as  it  travels  along.  The 
effect  of  the  electrical  resistance  of  the  two  bounding  wires  (or 
metal  sheets)  is  to  cause  a  steady  decay  of  the  magnetic  field,  and 
the  effect  of  imperfect  insulation  of  the  material  between  the 
bounding  wires  is  to  cause  a  continual  decay  of  the  electric  field. 
The  continual  decay  of  the  magnetic  field  may  be  thought  of  as 
due  to  the  resistance  which  opposes  the  slipping  of  the  rotating 
ether  cells  where  they  are  in  contact  with  the  bounding  wires  in 
Fig.  195,  and  the  continual  decay  of  the  electric  field  is  somewhat 
analogous  to  the  slow  disappearance  of  stress  in  a  stretched  piece 
of  rubber  which  may  be  supposed  to  have,  in  addition  to  its  elastic 
property,  a  certain  degree  of  viscosity,  like  pitch,  so  as  to  con- 
tinually yield  under  the  influence  of  the  stress.  When  the  re- 
sistance per  unit  length  of  the  bounding  wires  in  Fig.  195  bears 
a  certain  ratio  *  to  the  insulation  resistance  of  the  material  be- 
tween unit  length  of  the  bounding  wires,  then  the  electric  and 
magnetic  fields  decay  in  such  a  way  as  to  continually  satisfy  equa- 
tion (79),  and  the  wave  progresses  without  changing  its  shape.  A 
pair  of  transmission  wires  which  satisfies  this  condition  constitutes 
what  is  called  a  distortionless  line.  In  all  ordinary  telephone  lines 
the  effect  of  line  resistance  is  greatly  in  excess  of  the  effect  of  line 
leakage,  f  and  therefore  an  electric  wave  in  being  transmitted  along 

*  This  relation  may  be  quite  easily  formulated  but  an  elaborate  discussion  of  wave- 
distortion  is  not  within  the  scope  of  this  text. 

t  Several  interesting  examples  are  given  by  B.  S.  Cohen  in  The  Electrician 
(London),  April  lo,  1908. 


270        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


•mood 


a  telephone  line  suffers  continual  distortion  because  of  the  rapid 
decay  of  magnetic  field  due  to  line  resistance.  The  distortion  of 
an  electric  wave  as  it  travels  along  a  pair  of  telephone  lines  is 
similar  in  many  respects  to  the  distortion  of  a  canal  wave  as  de- 
scribed in  Art.  144  and  as  represented  in  Fig.  194.  Imagine  a 
rectangular  electromagnetic  wave-pulse  to  be  started  at  the  middle 
of  a  long  telephone  line  (two  wires  of  course).  Let  the  small 
rectangle  in  the  upper  part  of  Fig.  194  represent  the  initial  form 
of  the  wave.  After  the  elapse  of  time  the  wave  changes  to  the 
shape  shown  by  BB,  Fig.  1 94.  The  energy  in  the  head  of  the 
wave  decreases  partly  because  of  the  RP  losses  in  the  line  wires 
and  partly  because  of  the  shooting  of  energy  back  into  the  tail 
of  the  wave. 

The  transmission  of  articulate  speech  over  a  telephone  line  de- 
pends upon  the  transmission  of  characteristic  shapes  of  electric 

waves.  Thus,  the  shapes  of  the 
electric  waves  necessary  to  re- 
produce certain  vowel  sounds  are 
shown  in  Fig.  202,  and  the  wave 
shapes  which  are  necessary  to 
produce  consonant  sounds  are 
very  much  more  complicated  than 
these.  The  wave  distortion  on 
the  line  tends  to  make  each  ele- 
mentary portion  of  a  wave  spread 
out  as  shown  in  Fig.  194,  and  if 
each  elementary  portion  of  a  com- 
plicated wave  spreads  out  in  this 
way  the  fine  details  of  wave  shape 
are  very  soon  obliterated  as  the  wave  travels  along. 

It  is  not  desirable  to  eliminate  wave  distortion  by  providing 
poor  insulation  between  telephone  wires  because  this  results  in  a 
great  reduction  in  the  amount  of  energy  transmitted.  The  method 
which  is  used  in  practice  is  to  connect  small  inductance  coils  in 
circuit  with  the  line  wires  at  intervals  over  the  whole  length  of 


fdr 


Fig.  202. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES. 

the  line.  The  effect  of  these  inductance  coils  is  to  permit  of  the 
satisfying  of  equation  (79)  (magnetic  energy  equal  to  electric 
energy)  with  a  very  greatly  reduced  value  of  current  in  the  line 
wires  so  that  the  RI2  loss,  which  is  the  cause  of  the  wave  dis- 
tortion, is  very  greatly  reduced.  A  telephone  line  provided  with 
inductance  coils  in  this  way  is  called  a  loaded  line.  This  arrange- 
ment is  due  to  Pupin. 

The  canal  analogue  of  a  loaded  telephone  line  is  as  follows  : 
Imagine  a  great  number  of  thin  boards  to  be  placed  across  the 
canal  in  the  form  of  diaphragms  but  free  to  move  with  the  water 
in  the  canal,  and  imagine  these  thin  boards  to  be  very  massive. 
The  effect  of  t'hese  massive  boards  would  be  to  reduce  the 
velocity  v  in  Fig.  1 90  and  still  permit  the  kinetic  energy  of  the 
moving  water  and  boards  to  be  equal  to  the  potential  energy  due 
to  the  elevation  of  the  water  in  the  wave.  This  reduced  velocity 
of  flow  v  would  greatly  reduce  the  friction  of  the  water  against 
the  sides  of  the  canal  and  therefore  the  kinetic  energy  of  the 
wave  would  be  dissipated  much  less  rapidly  than  if  the  water  in 
the  canal  were  not  loaded. 

The  loading  of  a  telephone  line  is  helpful  only  when  the  energy 
loss  due  to  line  resistance  is  much  greater  than  the  energy  loss 
due  to  line  leakage  (poor  insulation).  When  line  leakage  (poor 
insulation)  is  excessive,  the  loading  of  the  line  tends  to  increase 
wave  distortion.  The  explanation  of  this  effect  of  loading  is  as 
follows :  The  velocity  of  transmission  of  the  waves  along  a  line 
is  greatly  reduced  by  loading  so  that  a  longer  time  is  required 
for  a  wave  to  travel  over  the  line  and  therefore  the  wave  loses 
energy  by  leakage  for  a  longer  time.* 

PROBLEMS. 

143.  Ten  horse-power  is  transmitted  along  a  row  of  gear  wheels, 
the  speed  of  each  of  which  is  1,200  revolutions  per  minute.  The 

*  The  student  who  wishes  to  pursue  the  study  of  the  theory  of  electric  waves 
should  read  Heaviside's  Electromagnetic  Theory,  Vols.  I  and  II,  London,  The  Elec- 
trician Company.  The  second  part  of  Vol.  I,  namely,  pages  306  to  455*  is  especially 
instructive. 


2/2        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

diameter  of  the  pitch  circle  of  each  gear  is  2  feet.  Find  :  (a) 
The  tangential  force  exerted  on  a  given  gear  wheel  by  each  ad- 
jacent wheel,  and  (b}.  the  torque  exerted  upon  a  given  gear  by 
each  adjacent  gear  and  express'*  the  result  in  pound-feet.  Ans. 
(a)  2 1 J  pounds.  (fr)  2 1 J  pound-feet. 

144.  A  long  wire  of  which  the  resistance,  per  centimeter  of 
length  is  0.02  ohm  carries  a  current  of  3  amperes,     (a)  Find  the 
rate  at  which  energy  flows  in  upon  each  centimeter  of  length  of 
this    wire   in    ergs  per   second,     (ft)  Find  the  intensity    of  the 
energy  stream  at  a  distance  of  1 5  centimeters  from  the  wire  in 
ergs  per  second  per  square  centimeter,     (c)  Find  the  intensity  of 
the  electric  field  parallel  to  the  wire  in  abvolts  per  centimeter  and 
find  the  intensity  of  the  magnetic  field  in  gausses  at  a  distance  of 
15  centimeters  from  the  wire,     (d)  Find  the  value  of  the  pro- 
portionality factor  by  which  the  product  of  intensities  of  electric 
and  magnetic  fields  (at  right  angles  to  each  other)  must  be  mul- 
tiplied  to  give  the  intensity   of  an   energy   stream   in   ergs   per 
second   per    square   centimeter.     Ans.   (a)    1,800,000    ergs    per 
second,     (b)    19,100  ergs  per  second  per  square  centimeter.     (<r) 
6,000,000  abvolts  per  centimeter,     (d)    1/471-. 

145.  Consider  two  line  wires  in  the  form  of  two  flat  metal  rib- 
bons 50  centimeters  wide  and   3   centimeters  apart.     At  a  given 
point  pp  the  electromotive  force  between  the  ribbons  is  100  volts 


volis 


Generator  -< 

end  io  amperes 


Fig.  203. 

and  the  current  in  each  ribbon  is  io  amperes,  as  shown  in  Fig. 
203.  (a)  Find  the  rate  in  ergs  per  second  at  which  energy  flows 
past  the  given  point  /  from  generator  towards  receiver  in 


ELECTRIC   OSCILLATIONS   AND   ELECTRIC   WAVES.     273 

ergs  per  second,  using  the  ordinary  formula  P—  El.  (b) 
The  electric  field  intensity  between  the  points  //  in  Fig. 
203  is  33^  volts  per  centimeter  and  the  magnetic  field  between 
the  strips  is  uniform  and  perpendicular  to  the  plane  of  the  paper. 
Let  the  intensity  of  the  magnetic  field  be  H.  Express  the 
intensity  of  the  energy  stream  across  //  in  ergs  per  square 
centimeter  per  second  in  terms  of  H  and  the  electric  field, 
intensity  using  the  proportionality  factor  found  in  problem  147. 
Multiply  this  intensity  of  the  energy  stream  by  the  sectional  area 
across  which  it  flows  at  //  and  place  this  result  equal  to  El 
(expressed  in  c.g.s.  units  of  course)  and  thus  find  the  intensity  of 
the  uniform  magnetic  field  between  the  two  ribbons.  Ans.  (a) 
io10  ergs  per  second,  (fr)  4^/50  gauss. 

146.  A  water  wave  travels  along  a  canal  in  which  the  normal 
depth  of  water  is  6  feet,  the  width  of  the  canal  being  12  feet. 
The  wave  is  30  feet  long  and  the  water  in  the  wave  has  a  uniform 
velocity  of  0.3  foot  per  second.     Find  the  total  energy  of  the 
wave  counting  both  potential  energy  and  kinetic  energy.     Ans. 
379.7  foot-pounds. 

147.  A  rectangular  electromagnetic  wave -pulse  is  bounded  by 
two  broad  sheets  of  metal  as  shown  in  Fig.  195.     The  width  of 
the  sheets  is  50  centimeters,  their  distance  apart  is  3  centimeters 
and  the  length  of  the  wave  pulse  is  I  oo  centimeters.     The  inten- 
sity of  the  uniform  magnetic  field  in  the  region  of  the  wave  is  io 
gausses.     Find  the  total  energy  of  the  wave  including  electric 
and  magnetic  energy.     Ans.  3,000,000/877  ergs. 

148.  A  battery  of  which  the  electromotive  force  is  1,000  volts 
is  connected  at  a  given  instant  to  the  two  wires  of  a  transmission 
line.     Treating  the  transmission  line  as  though  it  consisted  of  two 
flat  ribbons,  make  a  diagram  somewhat  similar  to  Figs.  195  and 
196  showing  the  distribution  of  current  in  the  bounding  metal 
sheets,  the  distribution  of  charge  on  the  bounding  metal  sheets, 
and  the  distribution  of  electric  and  magnetic  fields  in  the  region 
between  the  sheets  at  an  instant    t   seconds  after  the  battery  is 
connected,    Vt  being  less  than  the  length  of  the  line,  where    V 


2/4        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

is  the  velocity  of  progression  of  electric  waves  along  the  line. 
Assume  in  this  problem  that  the  resistance  of  the  bounding  metal 
sheets  is  negligible.  --  , 

Note.  —  A  wave  of  starting  travels  out  from  the  battery  end  of  the  line  at 
velocity  V.  Ahead  of  this  wave  of  starting  the  line  is  wholly  undisturbed.  Behind 
this  wave  of  starting  the  current  in  each  line  has  everywhere  the  same  value  i  (out- 
wards in  one  line,  backwards  in  the  other  line),  the  electromotive  force  between  mains 
is  everywhere  equal  to  1,000  volts,  the  electric  field  intensity  between  the  ribbons 
has  everywhere  the  same  value,  and  the  magnetic  field  between  the  ribbons  has 
everywhere  the  same  value  H.  The  electric  energy  per  unit  length  of  the  pair  of 
ribbons  may  be  calculated  with  the  help  of  equations  (65^)  and  (62)  by  considering 
the  ribbons  as  the  two  plates  of  a  condenser,  the  electromotive  force  between  them 
being  1,000  volts.  The  intensity  of  the  magnetic  field  between  the  ribbons  may  then 
be  found  from  the  fact  that  the  electric  energy  must  be  equal  to  the  magnetic  energy, 
and  the  current  in  each  ribbon  may  then  be  found  from  the  relationship  established  in 
problem  147.  Calculate  the  intensity  of  the  electric  field,  the  intensity  of  the  mag- 
netic field,  and  the  current  in  each  ribbon  in  the  region  behind  the  wave  of  starting 
in  problem  148. 

149.  The  end  of  the  transmission  line  (pair  of  ribbons)  in  prob- 
lem 148  is  short-circuited  by  zero  resistance.      Make  a  diagram 
showing  the  distribution  of  electric  and  magnetic  field,  the  distri- 
bution of  charge  on  the  two  ribbons  and  the  distribution  of  cur- 
rent along  the  two  ribbons  at  an  instant 'after  the  wave  of  starting 
has  been  reflected  from  the  short-circuited  end  of  the  line. 

Note.  —  The  student  should  read  Art.  136  of  Franklin  and  MacNutt's  Elements 
of  Mechanics  in  order  to  be  able  to  understand  this  problem. 

150.  The   end  of  the   transmission  line  (pair  of  ribbons)  in 
problem  148  is  open,  that  is  the  two  ribbons  come  to  an  end  in 
air.     Make  a  diagram  showing  the  distribution  of  electric  and 
magnetic  field,  the  distribution  of  charge  on  the  two  ribbons  and 
the  distribution  of  current  along  the  two  ribbons  at  an  instant 
after  the  wave  of  starting   has   been    reflected   from  the  open 
end  of  the  line. 

151.  The  transmission  line  specified  in  problems  148,  149  and 
150  is  assumed  to  have  zero  resistance,  and  the  short-circuit  at 
the  end  of  the  line  is  assumed  to  have  zero  resistance  in  problem 
149  so  that  the  current  produced  by  the  battery  in  problem  149 
ultimately  becomes  indefinitely  large.     Plot  a  curve  showing  the 
growth  of  current  at  the  battery  terminals  with  lapse  of  time. 


ELECTRIC   OSCILLATIONS   AND    ELECTRIC   WAVES.     2/5 

Note, —  The  battery  current  starts  at  a  definite  value  t,  as  explained  in  the  note 
to  problem  148,  and  retains  this  value  until  the  wave  of  starting  travels  to  the  end  of 
the  line  and  back,  when  the  current  suddenly  increases  to  the  value  of  2t  and  so  on. 
The  effect  of  the  resistance  of  the  transmission  line  is  too  complicated  to  permit  of  its 
being  easily  taken  into  account,  and  therefore  the  resistance  of  the  transmission  line  is 
assumed  to  be  zero  in  problems  148,  149,  150  and  151. 

152.  A  long  train  of  cars  has  highly  elastic  springs  in  the 
couplers.  Describe  the  precise  manner  in  which  the  train  gains 
velocity  under  a  constant  pull  of  the  locomotive,  ignoring  friction. 

Note.  —  The  manner  of  starting  of  the  train  is  precisely  analogous  to  the  manner 
of  setting  up  a  current  in  the  transmission  line  in  problem  149. 


CHAPTER  X. 
ELECTRICAL  MEASUREMENTS. 

148.  Absolute  measurements   and   international  standards.  — 

The  measurement  of  an  electrical  quantity  in  terms  of  the  me- 
chanical units  of  length,  mass  and  time  directly  is  called  "  abso- 
lute "  electrical  measurement.  For  example,  the  measurement 
of  current  by  the  Weber  electro-dynamometer  as  explained  in 
Art.  59,  is  an  "  absolute "  measurement.  Absolute  electrical 
measurement  requires,  in  most  cases,  elaborate  apparatus,  and, 
unless  extreme  precautions  are  taken,  is  subject  to  considerable 
error.  In  consequence  of  this  fact  a  standard  of  resistance  and 
the  electrochemical  equivalent  of  silver  have  been  measured  "  ab- 
solutely "  with  extreme  care  and  adopted  as  international  stand- 
ards,* and  all  ordinary  electrical  measurements  consist  in  the 
comparison  of  the  quantity  to  be  measured  with  these  standards. 

MEASUREMENT  OF  CURRENT. 

149.  Measurement   of   current  by  electrolysis. —  The  electro- 
chemical equivalent  of  a  metal  having  been  determined  once  for 
all,  the  strength  of  any  current   may  be   easily  and  accurately 

*The  international  standard  ampere  is  defined  in  Art.  3,  and  the  method  by  which 
it  was  determined  is  described  in  Art.  59.  The  international  standard  ohm  is  defined 
in  Art.  52,  and  the  method  by  which  it  was  determined  is  described  in  Art.  152.  It 
is  likely  that  the  electromotive  force  of  the  Clark  standard  cell  (see  Art.  159)  will  be 
adopted  as  an  international  standard  at  the  next  International  Electrical  Congress. 
In  fact,  all  practical  electrical  measurements  are  now  based  upon  the  standard  cell 
and  a  standard  ohm.  The  use  of  the  silver  voltameter  is  very  tedious  and  the  results 
obtained  are  less  reliable  than  those  which  may  be  obtained  with  great  ease  by  the 
use  of  a  standard  ohm  and  a  standard  cell. 

An  historical  sketch  of  the  international  units  by  Frank  A.  Wolff  is  to  be  found  in 
the  Bulletin  of  the  United  States  Bureau  of  Standards,  Vol.  I,  pages  39-76.  The 
Acts  of  Congress  establishing  the  legal  electrical  units  for  the  United  States  are  given 
on  pages  61-65. 

276 


ELECTRICAL   MEASUREMENTS.  277 

measured  by  weighing  the  metal  deposited  by  the  current  during 
an  observed  interval  of  time. 

An  electrolytic  cell  arranged  for  the  measurement  of  current 
by  electrolysis  is  called  a  coulombmeter.  Thus  we  have  the  silver 
coulombmeter  (which  is  described  on  page  21),  the  copper  cou- 
lombmeter, and  the  water  coulombmeter.  The  water  coulombmeter 
consists  of  an  electrolytic  cell  with  platinum  electrodes  and  con- 
taining dilute  sulphuric  acid.  It  is  arranged  so  that  the  liberated 
oxygen  and  hydrogen  may  be  collected  and  its  volume  measured. 

150,  Measurement  of  current  by  the  potentiometer  and  a  stand- 
ard resistance.*  —  The  most  convenient  method  for  measuring 
current  accurately  in  the  laboratory  is  to  send  the  current  through 
a  standard  resistance  and  measure  the  electromotive  force  across 
the  terminals  of  the  resistance  by  means  of  a  potentiometer,  as 
explained  in  Art.  159.     This  method  is  convenient  because  it  is 
very  much  quicker  than  the  electrolytic  method  and  it  is  quite 
accurate  because  standard  resistances  are  now  available  which  are 
reliable  to  within,  say,  o.oi  of  one  per  cent,  and  the  electromo- 
tive force  across  the  resistances  can  be  measured  by  means  of  the 
potentiometer  in  terms  of  the  accurately-known  electromotive 
force  of  the  standard  cell. 

151.  Direct-reading  ammeters.  —  An  ammeter  is  a  galvanom- 
eter with  a  pointer  which  plays  over  a  scale  which  is  divided  and 
numbered  so  that  the  reading  of  the  pointer  gives  the  value  of  the 
current  directly.     The  ammeter  which  is  described  in  Art.  I  con- 
sists of  a  pivoted  coil  through  which   the  current  flows,  and  a 
permanent  magnet  which  deflects  the  coil.     This  arrangement  is 
essentially  similar  to  the  D'Arsonval  galvanometer  which  is  de- 
scribed in  Art.  61.     Another  type  of  ammeter,  the  electrodyna- 
mometer  type  (see  Art.  59),  is  used  generally  for  alternating-cur- 
rent measurements.      It  consists  of  a  pivoted  coil  and  a  fixed  coil 
connected  in  series.     The  current  to  be  measured  flows  through 
both  coils  and  the  force  action  between  the  coils  causes  the  pivoted 

*  See  Practical  Physics,  by  Franklin,  Crawford  and  MacNutt,  pages  62—74. 


278        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

coil  to  be  deflected.  The  plunger  type  of  ammeter  is  extensively 
used  where  cheapness  is  a  prime  consideration.  In  instruments 
of  this  type  the  current  flows  through  a  coil  of  wire  which  mag- 
netizes and  moves  a  pivoted  or  suspended  piece  of  soft  iron  to 
which  the  pointer  is  attached. 

MEASUREMENT  OF  RESISTANCE. 

152.  Absolute  measurement  of  resistance.    Lorenz's  method. —  A  circular  disk 
of  copper  DDt    Fig.  204,  is  mounted  on  an  axle  and  driven  at  a  uniform  speed  of  n 


s 

»•••• 


pulley 

axle  . .  _,_  B 


#  3.-  - 

Fig.  204. 

revolutions  per  second.  This  disk  is  surrounded  by  a  large  coil,  or  solenoid,  SS 
through  which  a  steady  current  /  (the  value  of  which  need  not  be  known),  flows 
from  a  battery  B.  This  current  also  flows  through  the  resistance  R  which  is  to  be 
measured,  and  an  auxiliary  circuit  containing  a  sensitive  galvanometer  G  is  connected 
so  that  the  electromotive  force  which  is  induced  in  the  rotating  disk  between  the 
brushes  a  and  b  can  be  balanced  against  the  electromotive  force  RI  across  the  re- 
sistance R,  this  balance  being  indicated  by  zero  deflection  of  the  galvanometer.  The 
intensity  of  the  magnetic  field  in  the  solenoid  SS  is  H=$TrzI,  everything  being 
expressed  in  c.g.s.  units  and  z  being  the  number  of  turns  of  wire  per  centimeter 
length  of  the  solenoid.  Any  given  radial  filament  of  the  rotating  disk  cuts  Trr2  X  H 
lines  of  flux  during  each  revolution  of  the  disk,  so  that  the  electromotive  force  induced 
between  the  center  and  the  circumference  of  the  disk  is  equal  to  7rr*X-^Xw  or 
Trr2  X  47r2^X  n'  When  the  speed  of  the  disk  is  increased  until  the  galvanometer 
gives  no  deflection,  then  this  induced  electromotive  force  is  equal  to  RI,  whence  we 
have 

RI—  Trr2  X  4'^/X  n 
or 

R  = 


153.  Resistance  boxes.  —  The  measurement  of  resistance  ordi- 
narily consists  of  the  determination  of  a  given  resistance  in  terms 
of  a  known  resistance.  In  many  cases  this  measurement  is  ac- 
complished by  adjusting  a  known  resistance  until  its  equal  to  the 
resistance  to  be  measured,  and  a  resistance  box  is  an  arrangement 


ELECTRICAL   MEASUREMENTS. 


279 


by  means  of  which  any  desired  known  resistance  may  be  intro- 
duced into  a  circuit.  The  usual  construction  of  the  resistance  box 
is  as  follows  :  A  series  of  massive  metal  blocks  are  connected  by 
wires  whose  resistances  are  I,  2,  2,  5,  10,  10,  20,  50  ohms,  etc., 
respectively.  By  means  of  conical  metal  plugs  which  fit  snugly 
between  the  blocks,  the  blocks  may  be  connected  at  pleasure, 
leaving  the  resistance  between  them  approximately  equal  to  zero. 
Figure  206  shows  the  essential  features  of  this  construction. 


Fig.  205. 


Fig.  206. 


154.  Measurement   of    resistance   by  Wheatstone's   bridge. — 

Wheatstone's  bridge  consists  of  a  net-work  of  conductors,  as 
shown  in  Fig.  206.  A  battery  circuit  branches  at  the  points  a 
and  3,  and  the  current  flows  through  the  four  resistances  a,  /£, 
7  and  8,  as  shown.  A  sensitive  galvanometer  G  is  connected 
between  the  points  c  and  d.  When  no  current  flows  through 
the  galvanometer  the  four  resistances  a,  ft,  7  and  8  satisfy  the 
equation 

H 

The  method  of  using  this  arrangement  for  the  measurement  of 
current  is  explained  in  Arts.  155  and  156. 

Proof  of  equation  (82). —  Let  if  be  the  current  flowing  through 
a  and  ft  (the  same  current  flows  through  a  and  ft,  since  the 
galvanometer  current  is  zero)  and  let  i"  be  the  current  flowing 
through  7  and  S.  Inasmuch  as  there  is  no  current  flowing 


280        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

through  the  galvanometer  the  electromotive  force  between  c  and 
d  must  be  equal  to  zero.  Therefore  the  electromotive  force  ai' 
between  a  and  c  is  equal  to  the  electromotive  force  yi"  be- 
tween a  and  d,  that  is, 


0.1' 


and  similarly  we  find 


Dividing  equation  (i)  by  equation  (ii),  member  by  member,  we 
have  equation  (82). 

155.  Measurement  of  resistance  by  the  slide  wire  bridge.  —  A 
stretched    wire    abt    Fig.    207,    an    unknown    resistance    a,    a 


•MHW 

B 

Fig.  207. 


known  resistance  /3,  and  a  sensitive  galvanometer  G  are  con- 
nected as  shown  to  a  battery  B.  The  lettering  in  Fig.  207  cor- 
responds to  that  in  Fig.  206.  The  sliding  contact  d  is  adjusted 
until  the  galvanometer  gives  no  deflection  and  then  equation  (82) 
is  satisfied,  but  7/8  is  equal  to  the  ratio  of  the  lengths  of  the 
corresponding  portions  of  the  wire  ab,  and  it  is  easily  deter- 
mined by  measuring  the  lengths  ad  and  db.  Therefore,  ft 
being  known,  a  may  be  calculated. 

156.  Measurement  of  resistance  by  the  box  bridge.  —  The  box 
bridge  is  a  resistance  box  containing  three  sets  of  resistances, 
/3,  7  and  8  connected  as  shown  in  Fig.  208.  The  dotted  lines 
represent  connections  outside  the  box.  The  portions  7  and  B 
usually  have  each  a  lo-ohm,  a  loo-ohm,  and  a  i,ooo-ohm  coil, 


ELECTRICAL   MEASUREMENTS. 


28l 


so  that  the  ratio  7/8  may  have  a  series  of  values  any  one  of 
which  may  be  chosen  at  will.  The  portion  0  contains  usually 
I,  2,  2,  and  5  of  each  units,  tens,  hundreds,  etc.,  of  ohms.  An 
unknown  resistance  a  is  connected  as  shown,  the  ratio  7/8  is 
chosen,  and  the  value  of  0  is  changed  until  the  galvanometer 


-Htmtl 


Fig.  208. 

gives  no  deflection,  the  battery  key  K'  being  closed  first  and 
the  galvanometer  key  K  afterwards.*  The  value  of  a  is  then 
calculated  with  the  help  of  equation  (82). 

157.  The  measurement  of  resistance  by  the  ammeter  and  volt- 
meter. —  In  the  dynamo  testing  laboratory,  where  it  is  usually 
inconvenient  to  use  Wheatstone's  bridge,  resistance  is  ordinarily 
measured  by  means  of  an  ammeter  and  a  voltmeter  as  follows  :  A 
current,  which  is  measured  by  an  ammeter,  is  sent  through  the 
resistance,  and  the  electromotive  force  between  the  terminals  of 
the  resistance  is  measured  by  means  of  a  voltmeter.  The  value 

*  If  the  galvanometer  circuit  is  closed  when  the  battery  key  Kf  is  closed,  a  moment- 
ary pulse  of  current  may  flow  through  the  galvanometer  even  if  equation  (82)  is  satis- 
fled.  In  order  that  this  momentary  pulse  of  current  may  not  flow,  a  certain  relation  must 
exist  between  the  inductances  of  the  four  arms  of  the  bridge  a,  /?,  y  and  A.  This 
pulse  of  current  due  to  inductance  is  made  use  of  in  Maxwell's  method  of  measuring 
inductance  by  means  of  the  Wheatstone's  bridge.  See  Practical  Physics,  Franklin, 
Crawford  and  MacNutt,  Vol.  II,  pages  129-133. 


282        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

of  the  resistance  is  then  found  by  dividing  the  electromotive  force 
by  the  current. 

158.  Measurement  of   very  h|gh   resistances.      Insulation   re- 
sistance. —  Very  large  resistances  cannot  be  easily  measured  by 
the  methods  outlined  above.     Consider,  for  example,  an  insulated 
cable  consisting  of  a  core  of  copper  wire  surrounded  by  a  layer 
of  rubber  and  inclosed  in  a  sheath  of  lead.      If  the  lead  sheath 
is  connected  to  one  terminal  of  a  battery  and  the  copper  core  to 
the  other  terminal,  a  certain  amount  of  current  will  flow  through 
the  insulating  layer  of  rubber,  that  is  to  say,  the  rubber  is  not  a 
perfect  insulator  (infinite  resistance).     Very  high  resistances  are 
usually  determined  by  measuring,  with  a  sensitive  galvanometer, 
the  current   /  which  is  forced  through  the  given  resistance  by  a 
large  known  electromotive  force    E.     Then  according  to  Ohm's 
Law  the  resistance  is  equal  to    E/f.* 

Example.  —  One  terminal  of  a  1 ,000 -volt  battery  is  connected 
through  a  very  sensitive  galvanometer  to  the  outside  tin-foil 
coating  on  a  glass  jar,  and  the  other  terminal  of  the  battery  is 
connected  to  the  inside  coating.  The  current,  as  indicated  by  the 
steady  deflection  of  the  galvanometer,  is  1.4  x  io~10  amperes. 
The  resistance  of  the  glass  between  the  coatings  is  therefore 
equal  to  7,100,000  megohms  (one  megohm  is  equal  to  1,000,000 
ohms). 

MEASUREMENT  OF  ELECTROMOTIVE  FORCE. 

159,  The  potentiometer.  —  The  potentiometer  is  a  device  which 
is  now  extensively  used  for  the  accurate  measurement  of  electro- 
motive force.     The  essential  features  of  this  instrument  may  be 
best  described  by  referring  to  the  slide-wire  form  f  of  the  poten- 
tiometer, the  essential  features  of  which  are  shown  in  Fig.  209. 

*  Insulators  do  not  conform  to  Ohm's  Law,  or,  in  other  words,  the  current  through 
an  insulator  is  not  strictly  proportional  to  the  electromotive  force.  Different  values 
will  therefore  be  obtained  for  the  insulation  resistance  according  to  the  value  of  elec- 
tromotive force  used. 

f  Commercial  forms  of  the  potentiometer  lor  accurate  electromotive  force  measure- 
ments are  described  in  Practical  Physics  by  Franklin,  Crawford  and  MacNutt,  Vol. 
II,  pages  66-74. 


ELECTRICAL   MEASUREMENTS. 


283 


A  bare  German  silver  wire  WW  is  stretched  upon  a  board  and 
connected  to  a  battery  B  so  that  an  invariable  current  i  flows 
through  it.  A  side  circuit,  containing  a  sensitive  galvanometer 
G  and  a  voltaic  cell  of  which  the  electromotive  force  e  is  to  be 


w 


Fig.  209. 

measured,  is  connected  to  the  wire  WW  by  means  of  two  slid- 
ing contacts  a  and  b.  The  sliding  contact  b  is  adjusted  until 
the  galvanometer  gives  no  deflection,  then 

e  =  ri  (i) 

where  r  is  the  resistance  of  the  portion  db  of  the  German  silver 
wire.  The  voltaic  cell  e  is  replaced  by  a  standard  cell  of  which 
the  electromotive  force  e'  is  known,  and  the  sliding  contact  b 
is  again  adjusted  until  the  galvanometer  gives  no  deflection. 

Then 

e'  =  r'i  (ii) 

where  r'  is  the  resistance  of  the  portion  abf  of  the  German 
silver  wire.  Dividing  equation  (i)  by  equation  (ii),  member  by 

member,  we  have 

e       r  ..... 

-.=  -.  fin) 


The  ratio   rjr* ',    however,  is  equal  to  the  ratio  of  the  lengths  of 
the  respective  portions  of  the  wire    WW,  and  this  ratio  may  there- 


284        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

fore  be  determined  by  measuring  these  lengths,  so  that  the  ratio 
of  the  two  electromotive  forces  is  then  known. 

Standard  cells.  —  The  Clark  standard  cell  is  described  on  page 
1 6.  Its  electromotive  force  in;  volts  at  t°  C.  is  given  by  the 
equation 

E=  1.4292  —  o.ooi23(/  —  1 8)  —  o.oo<aoo7(V  —  i8)2 

The  Weston  cell  is  similar  in  every  respect  to  the  Clark  cell 
except  that  cadmium  amalgam  and  cadmium  sulphate  are  used 
instead  of  zinc  amalgam  and  zinc  sulphate.  The  electromotive 
force  in  volts  of  the  cadmium  cell  (with  concentrated  solution)  at 
t°  C.  is  given  by  the  equation 

E=  1.0187  —  0.00003  5  (/  —  1 8)  —  o.oooooo65(/—  i8)2 

MEASUREMENT  OF  POWER. 

160.  Measurement  of  power   by  means  of   the   ammeter  and 
voltmeter.  —  The  power  delivered  to  an  electrical  circuit  may  be 
calculated  from  the  equation    P=  El,    when  the  current   /  in 
the  circuit  and  the   electromotive  force    E  between  (across)  the 
terminals  of  the  circuit  have  been  measured.     This  method  is 
applicable  only  in  the  case  of  direct  currents,  that  is,  where  the 
current    /  and  the  electromotive  force   E  are  steady  in  value. 

161.  Measurement  by  means  of  the  wattmeter. — The   watt- 
meter is  a  special  form  of  electrodynamometer  the  connections  of 

which  are  shown  in  Fig.  210.  A 
fixed  coil  of  coarse  wire  B  is  con- 
nected in  series  with  the  receiving 
circuit  to  which  the  power  to  be 
measured  is  delivered,  and  a  sus- 
pended  or  pivoted  coil  A  of  fine  wire 
is  connected  across  the  supply  mains 
210  in  series  with  a  non-inductive  resist- 

ance R.     The  total  current  i  which 

delivered  to  the  receiving  circuit  flows  through  the  fixed  coil  B,  is 
a  current  which  is  proportional  to  the  supply  voltage    e   flows 


ELECTRICAL   MEASUREMENTS.  285 

through  the  pivoted  coil  A  (this  current  is  equal  to  e/R),  and 
the  force  action  between  the  two  coils  causes  the  coil  A  to  move 
and  carry  a  pointer  over  a  divided  scale.  The  force  action  be- 
tween the  two  coils  is  proportional  to  the  product  of  the  currents 
in  the  respective  coils,  that  is,  the  force  action  is  proportional  to 
ejR  x  i  or  proportional  to  d  since  R  is  constant.  But  ei 
is  the  power  delivered  to  the  receiving  circuit,  and  therefore, 
since  the  force  exerted  on  the  pointer  is  proportional  to  the 
delivered  power,  the  scale  over  which  the  pointer  plays  may  be 
divided  and  numbered  so  as  to  indicate  watts  of  power  directly. 
The  wattmeter  is  always  used  for  the  measurement  of  power 
delivered  by  an  alternator. 

USE  OF  THE  BALLISTIC  GALVANOMETER. 
162.  Measurement  of  electric  charge  and  of  magnetic  flux  by 
means  of  the  ballistic  galvanometer.  —  (a)  When  a  ballistic  gal- 
vanometer is  used  to  measure  the  discharge  q  from  a  condenser, 
the  charged  condenser  is  connected  to  the  galvanometer  termi- 
nals, and  the  throw  d  of  the  galvanometer  is  observed.  Then 

q  =  kd  (i) 

in  which  k  is  a  proportionality  factor  which  is  called  the  reduc- 
tion factor  of 'the  galvanometer.  This  reduction  factor  is  gener- 
ally determined  by  observing  the  throw  d  produced  by  a  known 
amount  of  charge  q.  Thus  a  condenser  of  known  capacity  C 
may  be  charged  by  a  known  electromotive  force  E  and  dis- 
charged through  the  galvanometer,  giving  q  =  EC  =  kd  from 
which  k  may  be  calculated. 

(b)  The  ballistic  galvanometer  is  frequently  used  to  measure 
what  is  called  the  impulse  value  of  the  momentary  electromotive 
force  which  is  induced  in  a  coil  of  wire  during  the  time  that  the 
magnetic  flux  through  the  coil  is  changing  by  a  certain  amount. 
Thus,  a  coil  containing  Z  turns  of  wire  is  placed  in  a  magnetic 
field  so  that  a  certain  amount  of  magnetic  flux  4>  *  passes  through 

*  This  flux    0   represents  the  flux  through  a  mean  turn  of  wire  on  the  coil. 


286        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  coil.  The  coil  is  connected  to  a  ballistic  galvanometer,  then 
quickly  removed  from  the  field,  and  the  galvanometer  throw  d  is 
observed.  This  throw  is  proportional  to  the  product  Z3>,  so 

that  we  may  write 

k'd  (ii) 


in  which  k'  is  a  constant  for  a  given  value  of  '.the  resistance  of  the 
galvanometer  circuit,  and  it  is  to  be  determined  by  observing  the 
throw  produced  by  a  known  value  of  ZQ.  If  the  resistance  of 
the  galvanometer  circuit  is  changed,  the  value  of  k'  is  altered. 

The  value  of  ZQ  in  the  above  discussion  is  the  impulse  value 
of  the  electromotive  force  which  is  induced  in  the  coil  of  wire 
during  the  time  that  it  is  being  withdrawn  from  the  magnetic 
field.  Let  /  be  the  short  interval  of  time  which  elapses  during 
the  movement  of  the  coil.  Then  the  flux  through  the  coil 
changes  from  4>  to  zero  during  t  seconds,  the  average  rate  of 
change  of  flux  is  <!>//,  the  average  value  of  the  electromotive 
force  which  is  induced  in  the  coil  is  Z<&jt,  and  the  product  of 
this  average  electromotive  force  and  the  time  is  equal  to  Z4>. 
The  product  of  the  average  value  of  the  electromotive  force  and  the 
time  during  which  the  electromotive  force  continues  to  act  is  called 
the  impulse  value  of  the  electromotive  force. 

163.  Measurement  of  capacity  *  —  The  simplest  method  of 
measuring  the  capacity  of  a  condenser  is  to  charge  the  condenser 
by  a  known  electromotive  force,  discharge  it  through  a  ballistic 
galvanometer  of  which  the  reduction  factor  k  is  known,  and  ob- 
serve the  deflection  d  which  is  produced.  Then  q  =  kd  =  CE 
from  which  C  may  be  calculated. 

*  The  most  accurate  method  for  measuring  the  capacity  of  a  condenser  is  to  use  a 
rapidly  rotating  commutator-device  arranged  to  charge  the  condenser  a  known  number 
of  times  per  second  from  a  battery  of  known  electromotive  force  and  discharge  the  con- 
denser the  same  number  of  times  per  second  through  an  ordinary  galvanometer,  the 
steady  deflection  of  which  measures  the  average  value  of  the  current.  The  most  accu- 
rate method  for  determining  the  ratio  of  the  capacities  of  two  condensers  is  by  means 
of  Wheatstone's  bridge,  as  described  in  Practical  Physics,  Franklin,  Crawford  and  Mac- 
Nutt,  Vol.  2,  page  133.  A  method  for  measuring  the  ratio  of  the  inductances  of  two 
coils  by  means  of  Wheatstone's  bridge  is  described  in  Practical  Physics,  Franklin, 
Crawford  and  MacNutt,  Vol.  2,  page  129. 


ELECTRICAL   MEASUREMENTS.  287 

164.  Measurement  of  magnetic  flux.  —  Consider  an  iron  rod 
through  which  a  certain  amount  of  magnetic  flux  passes,  due,  for 
example,  to  the  magnetizing  action  of  a  winding  of  wire  through 
which  a  current  is  flowing.     A  reversal  of  this  magnetizing  cur- 
rent produces  a  sudden  reversal  of  the  magnetic  flux    O    through 
the  rod,  so  that  the  total  change  of  flux  (from  -f  <I>  to  —  4>)  is 
equal  to  2<l>.     An   auxiliary   coil   having    Z    turns  of  wire  is 
placed  upon  the  iron  rod  and  connected  to  a  ballistic  galvanom- 
eter, and  the  throw  d  of  the  ballistic  galvanometer  is  observed  at 
the   instant  of  reversal   of  the  magnetizing  current.     Then  we 
have  2<&Z=  k'd,    inasmuch  as  the  product  of  the  change  of  flux 
2O  and  the  number  of  turns  of  wire  in  the  coil  gives  the  impulse 
value  of  the  electromotive  force,  and  this  is  equal  to    k'd.     The 
reduction  factor   kf    of  the  ballistic  galvanometer  being  known,* 
the  value  of  4>    can  be  easily  calculated. 

MEASUREMENT  OF  MAGNETIC  FIELDS. 

165.  Gauss's  method  for  measuring  the  horizontal  component  Hf 
of  the  earth's  magnetic  field,  and  for  measuring  the  magnetic 
moment  of  a  magnet.  —  This  method  involves  two  independent 
sets  of  observations,  the  first  set   being  made  with    a  certain 
arrangement  of  apparatus  and  the  second  set  being  made  with  a 
different  arrangement  of  apparatus,  as  follows  : 

First  arrangement.  —  A  large  magnet  is  suspended  horizontally 
at  the  place  where  Hf  is  to  be  determined,  set  vibrating  about 
the  vertical  axis  of  suspension  and  the  time  t  of  one  complete 
vibration  is  determined  by  observation.  Then  from  equation 
(23)  we  have 

ATT  i\.  ITTI  f\ 

2    =  m/a*  (0 

The  moment  of  inertia   K  of  the  magnet  is  to  be  determined 
from  the  measured  dimensions  and  weight  (in  grams)  of  the  bar. 

*A  method  for  determining  the  value  of  kf  is  described  on  page  18,  Vol.  II, 
Practical  Physics,  Franklin,  Crawford  and  MacNutt. 


288        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

Second  arrangement. — A  small  magnet  ns,  Fig.  21 1,  is  sus- 
pended at  the  place  which  was  occupied  by  the  large  magnet  in 
the  first  arrangement ;  this  small  magnet  being  free  to  turn,  points 
in  the  direction  of  the  magnetic  field  in  which  it  is  placed,  that  is, 
in  the  direction  of  H1 '.  The  large  magnet  used  in  the  first 


H 


fc 


N 


t 


IT 


Fig.  211. 

arrangement  is  now  placed  with  its  center  at  a  distance  d  due 
magnetic  east  or  west  of  the  small  magnet  ns  as  shown  in  Fig. 
211.  The  large  magnet  then  produces  at  the  small  magnet  a 
magnetic  field  h  which  is  at  right  angles  to  H',  and  the 
small  magnet  then  points  in  the  direction  of  the  resultant  of 
h  and  //',  having  turned  through  the  angle  <£  which  is 
observed. 

From  the  diagram,  Fig.  211,  we  have 

h 

tan  <p  =  -pp  (ii) 

From  equation  ( 1 7)  we  have  —  mj(d  —  J/)2  as  the  expression 
for  the  intensity  of  the  magnetic  field  at  ns  due  to  the  south  pole 
of  the  large  magnet ;  and  -f-  mj(d  -f  J/)2  for  the  field  intensity  at 
ns  due  to  the  north  pole ;  so  that 


(iii) 


This  equation  may  be  simplified  as  follows  :  Reduce  the  frac- 
tions mj(d  —  J/)2  and  mj(d  -f-  J/)2  to  a  common  denominator. 
We  then  have 


ELECTRICAL  MEASUREMENTS.  289 

h  —  2ml d 


(  *    /2Y 


Multiply  numerator  and  denominator  of  the  second  member  of 
this  equation  by  (  d2  -f  J2)2  and  we  have 

d 


h  =  2ml d.  - 


In  this  expression  /4/i6  may  be  dropped,  since  /  is  small 
compared  to  d,  and  /4  is  very  small  compared  to  d*.  There- 
fore 

'•* 


-  3?  (''4 


Substitute  this  simplified  value  of  h   in  equation  (ii),  and  we 

have 

2 


The  large  magnet  may  now  be  placed  nearer  to  ns  (Fig.  2  1  1  ), 
say  at  distance  dv  the  corresponding  angle  of  deflection  being 
<f>v  and  we  have 


The  uncertain  quantity  /,  which  is  the  distance  between  the 
poles  of  the  large  magnet,  may  be  eliminated  from  equation  (v) 
with  the  help  of  equation  (vi),  giving 

ml      d5  tan  <j>  —  d£  tan  <^L 
~ 


Observations  and  calculations.  —  The  quantity  /,  equation  (i), 
is  observed  and  K  is  calculated  from  the  measured  mass  and 
dimensions  of  the  large  magnet,  leaving  only  ml  and  Hf  un- 


20 


290        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

known  in  equation  (i).  The  quantities  d,  dl  ,  <f>  and  (f>l  in 
equation  (vii)  are  observed,  leaving  only  ;;//  and  Hr  unknown 
in  (vii).  Equations  (i)  and  (vii)  then  enable  the  calculation  of 
both  ml  and  Hf. 

If  it  is  desired  to  determine  the  strength  of  the  poles  of  the 
large  magnet,  the  quantity  /  may  be  approximately  measured, 
and  m  calculated. 

This  method  *  for  determining  ml  and  H  was  devised  by 
Gauss. 

166.  Measurement  of  magnetic  field  intensity  by  means  of  the 
tangent  galvanometer.  —  When  the  value  of  the  horizontal  com- 
ponent of  the  earth's  magnetic  field   H1    is  known,  the  tangent 
galvanometer  may  be  used  to  measure  the  value  of  the  current 
in  amperes  or  abamperes,  as  explained  in  Art.  57.     If  a  known 
current  (measured  by  a  copper   coulombmeter,  for  example)  is 
sent  through  a  tangent  galvanometer  and  the  deflection    <£    ob- 
served, then  the  value  of  H1    may  be  calculated,  the  number  of 
turns  of  wire    Z    and  the  mean  radius    r   of   the  coil   being 
known. 

167.  Measurement  of  magnetic  field  intensity  by  means  of  the 
bismuth  inductometer.  —  The  bismuth  inductometer  is  a  small 
resistance  coil  made  of  fine  bismuth  wire.      Its  resistance  varies 
with  the  intensity  of  the   magnetic   field   in  which   it  is  placed. 
The  relation  between  resistance  and  field  intensity  being  once  for 
all  determined,  the  intensity  of  any  field  may  be  found  by  meas- 
uring the  resistance  of  the  inductometer  when  it  is  placed  in  the 
field. 

168.  Kohlrausch's  method  for  the  simultaneous  absolute  measure- 
ment of  the  horizontal  component  of  the  earth's  magnetic  field  and 
of  current.  —  The  coil  of  a  tangent  galvanometer  is  suspended 
so  as  to  enable  the  measurement  of  the  torque    T  with  which 
the  earth's  horizontal  field  H1  acts  upon  it.     This  torque  is  given 

*  For  fuller  discussion  of  Gauss's  method,  see  A.  Gray,  Absolute  Measurements 
in  Electricity  and  Magnetism,  Vol.  II,  page  69. 


ELECTRICAL   MEASUREMENTS.  291 

by  equation  (40)  in  which    TrZir2    may  be  written  for   A,    giving 

At  the  same  time  the  deflection  <£  of  the  needle  of  the  galvanom- 
eter is  observed  so  that,  according  to  equation  (36^),  we  have 

/-S-ten*  (ii) 


The  mean  radius  r  and  number  of  turns  of  wire  Z  in  the  coil 
being  known,  and  T  and  $  being  observed,  these  two  equa- 
tions determine  the  values  of  both  /  and  Hf. 


APPENDIX   A. 
TERRESTRIAL  MAGNETISM.* 

1 .  The  earth  a  great  magnet.  —  The  tendency  of  a  compass 
needle  to  set  itself  in  a  particular  direction  at  a  given  place  on  the 
earth  was  at  a  very  early  date  attributed  to  some  action  of  the 
earth.  The  famous  Dr.  Gilbert,  Physician  in  Ordinary  to  Queen 
Elizabeth,  in  his  Latin  treatise  f  put  forward  the  important  idea 
that  the  earth  is  a  great  magnet,  so  that,  in  the  language  of 


''axis  of 
rotation 


Fig.  1. 

Faraday,  there  exists  a  magnetic  field  around  the  earth.     The 
general  character  of  the  earth's  magnetic  field  as  to  its  direction 

*  A  fairly  complete  discussion  of  terrestrial  magnetism  with  many  important  refer- 
ences is  given  in  Gray's  Treatise  on  Magnetism  and  Electricity,  Vol.  I,  pages  58-84, 
Macmillan  and  Company,  1898. 

\De  Magnete  magneticisque  corporibus.     Translated  into  English  about  1902  . 

292 


TERRESTRIAL   MAGNETISM. 


293 


and  intensity  at  various  points  on  the  earth  is  that  which  would 
be  produced  by  a  large  magnet  inside  of  the  earth  with  its  axis 
slightly  inclined  to  the  axis  of  rotation  of  the  earth,  as  shown  in 
Fig.  i. 

2.  The  compass.     Definition  of   declination.  —  The   compass 
needle  is  a  horizontal  magnet  which  is  free  to  turn  about  a  verti- 
cal axis.     The  direction  in  which  such  a  needle  points  at  a  given 
place  on  the  earth  is  called  the  magnetic  meridian  at  that  place, 
and  the  angle  between  the  magnetic  meridian  and  the  geographic 
meridian  is  called  the  declination  *  of  the  earth's  magnetic  field 
at  a  given  place. 

3.  The  dip  needle.     Definition  of  inclination.  —  The  needle  of 
a  compass  is  usually  weighed  at  one  end  to  make  it  lie  in  a  hori- 
zontal   plane.     A  steel   bar   which    is    magnetized   after   being 
accurately  balanced  on  a  horizontal  pivot  constitutes  a  dip  needle. 
When  the  horizontal  pivot  of  the  dip  needle  is  placed  at  right 
angles  to  the  magnetic  meridian,  the 

needle  points  in  the  actual  direction 
of  the  earth's  magnetic  field,  as 
shown  by  the  two  suspended  mag- 
nets ns  and  ns  in  Fig.  I,  and  the 
angle  of  inclination  of  the  needle  is 
called  the  inclination  or  dip  of  the 
earth's  magnetic  field  at  the  given 
place.  Figure  2  is  a  general  view  of 
a  dip  needle,  or  dip  circle,  as  it  is 
usually  called. 

4.  Magnetic  elements.  —  The   di- 
rection  and    intensity  of  the  earth's 

magnetic  field  at  a  place  is  completely  specified  when  the  decli- 
nation, the  inclination,  and  the  value  of  the  horizontal  component 

*  Sometimes  called  the  variation  of  the  compass.  This  word  variation,  how- 
ever, is  here  used  to  designate  the  changes  which  are  continually  taking  place  in  the 
earth's  magnetic  field. 


Fig.  2. 


294        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

are    given.     These   three  things  therefore  constitute    what   are 
called  the  magnetic  elements  at  a  given  place.* 

5.  Magnetic  maps.  —  Figure  '3  is  a  map  of  the  world  showing 
the  lines  of  equal  magnetic  declination  for  the  year  1905,  The 
heavy  black  curves  pass  through  the  regions  where  the  compass 
needle  points  true  north,  and  the  numbers  attached  to  the  fine- 


Fig.  3. 
Lines  of  equal  magnetic  declination. 

line  curves  are  the  values  of  the  declination.  Thus,  in  England 
the  compass  points  20°  to  the  east  of  north  and  at  San  Francisco 
the  compass  points  about  17°  west  of  north.  Figure  4  is  a  map 
of  the  world  showing  the  lines  of  equal  magnetic  dip  or  incli- 
nation for  the  year  1905.  Thus,  the  dip  needle  stands  in  a 
horizontal  position  at  all  places  on  the  heavy  curve  which  is 
marked  zero,  the  magnetic  dip  in  England  is  about  70°  (north 
pole  of  dip  needle  down),  and  the  dip  at  Cape  Town,  South 
Africa,  is  about  55°  (south  pole  of  dip  needle  down).  Figure  5 
is  a  map  of  the  world  showing  the  lines  of  equal  horizontal  inten- 

*The  methods  in  use  in  the  Magnetic  Observatory  at  Kew,  England,  for  determin- 
ing the  magnetic  elements  are  fully  described  in  Stewart  and  Gee,  Elementary  Practi- 
cal Physics,  Vol.  II,  pages  274-313. 


TERRESTRIAL   MAGNETISM. 


295 


sity.     Thus,  in  England  the  horizontal  intensity  is  about  0.17 
gauss,  and  in  Florida  the  horizontal  intensity  is  about  0.30  gauss. 


180     IuO     120     90 


180     1  0     120     90      60      30  30      00      90     120     150     1 


Fig.  4. 
Lines  of  equal  magnetic  inclination. 


30       0 


00      90      120     150 


ISO     120      90      CO      30      0      30      GO      90     120     15( 


Fig.  5. 
Lines  of  equal  horizontal  intensity. 

6.  Variations  of  the  earth's  magnetic  field.  —  Each  one  of  the 
magnetic  elements,  declination,  inclination,  and  horizontal  inten- 
sity, is  subject  to  variations  of  four  distinct  kinds,  as  follows : 


296 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


(a)  The  diurnal  variation.  —  Each  magnetic  element  is  subject 
to  a  daily  periodic  change.  This  is  called  the  diurnal  variation. 
Thus,  the  curves  in  Figs.  6a  and  6b  show  the  diurnal  variation  of 


2    3    4     5    6     7     8     9   io   ii"<y*i3  14.  *5  16  17    18   19  20  21  22  23  24 


2    3     4    3     6     7     8     9    io 


Fig.  6a. 

14     15     16    17    l8    19    2O   27     22    23-24 


Fig.  6b. 

the  magnetic  declination  at  the  United  States  Magnetic  Observa- 
tory at  Baldwin,  Kansas.  Figure  6a  shows  the  diurnal  variation 
in  mid-summer  and  Fig.  6b  shows  the  diurnal  variation  in  mid- 
winter. 

(&)   The  annual  variation.  —  Each  magnetic  element  is  subject 
to  an  annual  periodic  change  which  is  called  the  annual  variation. 


24    22     20    tg     16    14     12     TO.-.  8      6      4       2.     02      4      6.     8.    16  .  12 


67 

ep 
70 

72- 

73 
74 


540 


1800 


declination 

Fig.  7. 


TERRESTRIAL   MAGNETISM.  297 

(<r)  The  secular  variation.  —  Each  magnetic  element  is  subject 
to  a  slow  change  from  year  to  year.  This  is  called  the  secular 
variation.  Thus,  the  curve  in  Fig.  7  shows  the  secular  variation 
of  the  magnetic  declination  at  London  from  1540  to  1890. 

(d)  Magnetic  storms.  —  Each  magnetic  element  is  subject  to 
erratic  variations.  These  erratic  variations  occur  at  times  of  great 
disturbances  in  the  sun  as  indicated  by  sun-spot  activity,  and  also 
at  times  of  great  disturbances  in  the  earth  such  as  volcanic  erup- 
tions and  earthquakes,  and  they  are  called  magnetic  storms. 


APPENDIX   B. 


SHIP'S  MAGNETISM  AND  THE  COMPENSATION  OF  THE 
COMPASS.* 

7.  The  ship's  compass. —  The  style  of  ship's  compass  which  is 
now  almost  universally  used  is  that  which  is  due  to  Lord  Kelvin. 
The  card  of  this  compass  is  shown  in  Fig.  8.  The  points  of  the 

compass  and  the  circle  divi- 
sions are  printed  on  a  paper 
ring  to  which  is  attached  a 
light  rim  of  aluminum  which 
keeps  it  in  shape.  Radial 
threads  connect  the  ring  to  a 
central  disk  which  contains  a 
sapphire  cap  by  which  the 
compass  is  supported  on  an 
iridium  point.  Eight  small 
magnets  of  glass-hard  steel 
are  tied  to  the  radial  threads 
four  on  either  side  of  the 
jewel  cap,  as  shown  in  the  figure.  The  entire  weight  of  the  card^ 
including  the  magnetic  needles,  is  170^  grains,  and  this  extreme 
lightness  combined  with  the  relatively  large  moment  of  inertia  due 
to  the  distribution  of  the  mass,  insures  a  long  period  of  free  vibra- 
tion and  therefore  great  steadiness.  The  lightness  of  the  card  also 

*  A  good  discussion  of  this  subject  is  given  in  Gray's  Treatise  on  Magnetism  and 
Electricity,  Vol.  I,  pages  85-100,  Macmillan  and  Company,  1898.  For  full  details, 
the  reader  is  referred  to  Lord  Kelvin's  Instructions  for  Adjusting  the  Compass,  to  be 
obtained  from  James  White,  of  Glasgow.  The  practice  in  the  United  States  Navy 
concerning  the  matter  of  compass  errors  and  compass  adjustments  is  given  in  several 
small  pamphlets  which  are  published  by  the  United  States  Navy  Department,  and  in 
a  book  entitled  A  Treatise  on  Navigation,  by  Commander  W.  C.  P.  Muir,  U.  S. 
Navy,  Annapolis,  1906.  The  practice  in  the  British  Navy  is  given  in  the  Admiralty 
Manual  of  Deviations  of  the  Compass. 

298 


Fig.  8. 


SHIP'S   MAGNETISM.  299 

gives  a  very  small  frictional  resistance  at  the  supporting  point. 
The  compass  card  with  its  attached  needles  is  supported  in  a 
copper  bowl  which  is  supported  on  gimbals,  so  that  the  compass 
remains  horizontal  in  spite  of  the  rolling  motion  of  the  ship. 
The  complete  instrument  is  supported  on  a  column  which  con- 
tains or  supports  the  compensating  devices  which  are  explained 
later,  and  the  entire  arrangement  is  called  the  binnacle. 

When  a  ship  contains  no  iron  or  steel  the  compass  points  in 
the  direction  of  the  magnetic  meridian,  and  a  chart  like  Fig.  3 
enables  a  navigator  to  infer  the  true  heading  of  a  ship  from  an 
observed  reading  of  the  compass.  When,  however,  the  ship  is 
made  of  iron  or  steel,  or,  when  it  carries  a  cargo  of  iron  or  steel, 
the  compass  is  usually  deflected  by  the  magnetism  of  the  ship  or 
of  its  cargo.  In  order  that  a  compass  may  be  used  for  purposes 
of  navigation  under  such  conditions  the  errors  of  the  compass 
may  be  determined  by  a  careful  set  of  observations,  or  the  in- 
fluence of  the  ship's  magnetism  may  be  compensated,  thus 
reducing  the  compass  errors  approximately  to  zero.  The  latter 
method  is  the  one  which  is  usually  employed,  and  in  some  cases 
the  residual  errors  which  remain  on  account  of  incomplete  com- 
pensation are  determined  by  a  careful  set  of  observations  and 
allowed  for  in  the  use  of  the  compass. 

8,  Ship's  magnetism.  — A  ball  of  iron  which  is  devoid  of  per- 
manent magnetism,  is  weakly  magnetized  by  the  earth's  field. 
This  magnetism,  which  is  not  in  a  fixed  direction  in  the  ball,  but 
which  is  always  in  the  direction  of  the  earth's  field  however  the 
ball  may  be  held  or  turned,  is  called  the  temporary  magnetism 
of  the  ball,  and  it  is  proportional  to  the  intensity  of  the  earth's 
field.  If  the  ball  is  elongated  like  an  ellipsoid  its  temporary 
magnetism  is  not  in  general  parallel  to  the  earth's  field,  and  in 
the  case  of  a  long  slim  iron  rod  its  temporary  magnetism  is  in 
the  direction  of  its  length  and  proportional  to  the  component  of 
the  earth's  field  which  is  parallel  to  it  inasmuch  as  that  part  of 
the  earth's  field  which  is  at  right  angles  to  a  slim  rod  produces 
no  perceptible  magnetism. 


300        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

When  a  ball  or  rod  of  iron  has  a  certain  amount  of  permanent 
magnetism,  the  effect  of  the  earth's  magnetic  field  upon  it  is  to 
produce  an  additional  temporary  magnetism,  that  is  to  say,  the 
magnetism  of  the  ball  or  rod  is  the  sum  of  two  distinct  parts,  a 
permanent  magnetism  and  a  temporary  magnetism.  Of  course 
the  permanent  magnetism  of  a  rod  may  be  changed  by  severe 
mechanical  shocks ;  the  word  permanent  here  refers  to  that  part 
of  the  magnetism  which  does  not  change  as  the  ball  or  rod  is 
slowly  moved  around  in  the  earth's  field. 

Similarly,an  iron  ship  has  a  certain  amount  of  permanent  magnet- 
ism which  does  not  change  as  the  ship  moves  around  in  the  earth's 
magnetic  field  and  a  certain  amount  of  temporary  magnetism 
which  is  due  to  the  magnetizing  action  of  the  earth's  magnetic  field. 
9.  Compass  errors  due  to  permanent  magnetism  of  a  ship. — The 
permanent  magnetism  of  a  ship  produces  at  the  compass  box  a 
magnetic  field  which  is  constant  in  value  and  fixed  in  direction 
with  reference  to  the  ship.  The  horizontal  com  - 
ponent  of  this  field  combines  with  the  horizontal 
component  of  the  earth's  field  to  give  a  resultant 
field  in  the  direction  in  which  the  compass  needle 
points.  Thus,  Hf  in  Fig.  9  represents  the  hori- 
zontal component  of  the  earth's  field,  P  repre- 
sents the  horizontal  part  of  the  magnetic  field  at 
the  compass  which  is  due  to  the  permanent  mag- 
netism of  the  ship,  R  represents  the  resultant 
horizontal  field  at  the  compass,  and  6  represents 
the  compass  error  due  to  the  ship's  permanent  magnetism.  The 
field  P  rotates  with  the  ship  and  therefore  the  compass  error  0 
has  a  series  of  positive  values  (to  the  east)  throughout  a  half 
revolution  of  the  ship,  and  a  series  of  negative  values  (to  the 
west)  throughout  a  half  revolution  of  the  ship.  Therefore  the 
compass  error  due  to  the  ship's  permanent  magnetism  is  called 
the  semicircular  error* 

*The  permanent  magnetism  of  the  ship  contributes  also  to  the  heeling  error 
which  is  discussed  in  Art.  14,  and  the  so-called  semicircular  error  is  due  partly  to 
the  temporary  magnetism  of  the  ship  as  explained  in  Art.  13. 


SHIP'S   MAGNETISM.  301 

10.  The  semicircular  correctors.  —  The  ideal  compensation  for 
the  compass  errors  due  to  a  ship's  permanent  magnetism  would 
be  to  place  a  permanent  steel  magnet  in  such  a  position  that  it 
would  produce,  at  the  compass  box,  a  magnetic  field  equal  and 
opposite  to  the  field  produced  at  the  compass  box  by  the  ship's 
permanent  magnetism.     As  long  as  the  ship  remains  on  an  even 
keel,  however,  it  is  only  the  horizontal  part   Pt    Fig.   9,  of  the 
field  which  is  produced  at  the  compass  box  by  the  permanent 
magnetism  of  the  ship,  which  causes  the  deflection  of  the  compass. 
Therefore  it  is  sufficient  to  neutralize  this  horizontal  field    P.     For 
this  purpose,  one  or  more  horizontal  magnets  are  placed  in  trays 
in  the  pedestal  of  the  binnacle  and  adjusted  until  they  produce  a 
field  at  the  compass   box   which   is  equal   and   opposite   to    P. 
Usually,  two  such  trays  are  employed,  in  one  of  which,  magnets 
are  placed  parallel  to  the  line  of  the  keel  of  the  ship  so  as  to  annul 
the  bow  component  of  P,   and  in  the  other  of  which,  magnets  are 
placed  at  right  angles  to  the  line  of  the  keel  so  as  to  annul  the 
athwart-ship  component  of  P.     These  two  trays  with  their  per- 
manent magnets  are  called  the  semicircular  correctors. 

11.  Compass  errors  due  to  temporary  magnetism  of  a  ship.  — 
An  idea  of  the  general  character  of  the  compass  errors  which  are 
due  to  the  temporary  magnetism  of  a  ship  may  be  obtained  by 
imagining  the  ship  to  be  a  long  slim  bar   AB,    Fig.  10,  with  a 
compass  box  at  the  point    C.     The  earth's  horizontal  field  H' 
may  be  resolved  into  two  components,  one  parallel  to    AB   and 
the  other  at  right  angles  to   AB.     The  component  which  is  at 
right  angles  to    AB   has  no  perceptible  magnetizing  action  on 
AB,     the  component  which  is  parallel  to   AB   causes  the  end 
B   to    become  a  north  pole  and  the  end   A    to  become  a  south 
pole,  and  the  magnetic  field  at    C  due  to  these  magnet  poles  is 
parallel  to   BA    and  towards   A.     The   magnetic    field   at  the 
•compass  box  which  is  due  to  the  temporary  magnetism  of  the 
bar   AB   in  Fig.  i  o  is  represented  by  the  arrow    T  in  Fig.  1 1 , 
the  earth's  field  at  the  compass  box  is  represented  by  Hr y  and 
the  arrow   R    represents  the  resultant  field  at  the  compass  box 


302        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


in  the  direction  of  which  the  compass  needle  points ;  therefore 
the  angle  <£  is  the  compass  error.  Imagine  By  Fig.  10,  to  rep- 
resent the  bow  of  the  ship,  and  suppose  the  ship  to  start  with  its 
bow  due  north  and  swing  arouricl  to  the  east,  the  angle  a  increas- 
ing from  zero  to  360°.  A  careful  consideration  of  Figs.  10  and 
1 1  will  show  that  the  angle  <£  has  a  series  of  westerly  values 


H' 


Fig.  10. 


Fig.  11. 


throughout  the  first  quadrant  (a  between  zero  and  90°),  a  series 
of  easterly  values  throughout  the  second  quadrant  (a  between 
90°  and  1 80°),  a  second  series  of  westerly  values  throughout  the 
third  quadrant  (a  between  180°  and  270°),  and  a  second  series 
of  easterly  values  throughout  the  fourth  quadrant.  The  compass 
error  due  to  the  temporary  magnetism  of  a  ship  is  therefore  called 
the  quadrantal  error. 

When  the  ship's  compass  is  located  on  the  center  line  of  the 
ship  so  that  the  iron  of  the  ship  is  symmetrically  placed  on  the 
two  sides  of  the  compass,  then  the  compass  error  due  to  the 
ship's  temporary  magnetism  is  zero  when  the  ship  heads  north, 
east,  south,  or  west,  as  may  be  shown  as  follows  :  When  the  ship 
heads  magnetic  north  or  south,  its  temporary  magnetism  is  sym- 
metrical as  shown  in  Fig.  12,  the  magnetic  field  at  the  compass 
due  to  the  temporary  magnetism  of  the  ship  is  therefore  due 
south,  and  consequently  the  compass  is  not  deflected.  Figure 
13  shows  a  compass  box  C  placed  on  the  center  line  of  a  ship 
of  which  the  dissimilarity  of  bow  and  stern  is  greatly  exaggerated. 


SHIP'S    MAGNETISM. 


303 


Imagine  the  vessel  to  be  made  of  solid  iron  and  consider  the 
transverse  slice  of  iron  which  lies  between  the  dotted  lines  in  Fig. 


•A; 


s 


Fig.  12. 


13.  When  the  bow  points  east  or  west  (magnetic)  the  transverse 
slice  is  magnetized  as  indicated  by  the  letters  N  and  S,  and  the 
curved  line  Jf,  which  represents  a  line  of  force  due  to  the  poles 
N  and  5,  shows  that  the  field  at  C  which  is  produced  by  the 


stern 


Fig.  13. 


magnetism  of  the  transverse  slice  is  towards  the  south.     What  is 
here  said  concerning  a  given  transverse  slice  of  the  ship  is  true 


> 


304       ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

of  every  transverse  slice,  and  therefore  the  field  at  C  due  to  the 
transverse  magnetization  of  the  entire  ship  when  its  bow  points 
feast  or  west  (magnetic)  is  towards  the  south, 
and  consequently  the  compass  is  not  de- 
flected. 

12.  Compensation   of,  quadrantal     error. 

Quadrantal  correctors.  —  From  Figs.  1 2 
and  13  it  is  evident  that  the  temporary 
magnetism  of  the  ship  weakens  the  field  at 
the  compass  (T  opposite  to  H'  in  direction, 
so  that  the  resultant  of  T  and  H1  is  less 
than  H1 )  when  the  ship  heads  north,  east, 
south  or  west.  If  the  value  of  T  is  the 
same  in  Figs.  12  and  /j,  it  can  be  shown 
that  the  temporary  magnetism  of  the  ship 
does  not  tend  to  deflect  the  compass,  what- 
ever the  direction  of  the  bow  of  the  vessel. 
pj  To  prove  this  proposition,  we  will  assume 

that  the  iron  of  the  ship  is  equivalent  to 

two  long  slim  horizontal  bars  of  iron,  one  parallel  to  the  ship's 
keel  (the  A-bar)  and  the  other  at  right  angles  thereto  (the  B-bar). 


'B-bar 


-bow 


Fig.  15. 


When  the  ship  is  headed  north  as  shown  in  Fig.  14,  the  full  value 
of  fP    acts  to  magnetize  the  A-bar,  and  the  magnetic  field    T.9 


SHIP'S   MAGNETISM. 


305 


which  is  produced  at  the  compass  box  by  the  magnetization  of 
the  A-bar,  is  proportional  *  to  H'  or  equal  to  k^H' .  When  the 
ship  is  headed  east,  as  shown  in  Fig.  15,  the  full  value  of  Hf  acts 
to  magnetize  the  B-bar,  and  the  magnetic  field  Tv  which  is  pro- 
duced at  the  compass  box  by  the  magnetization  of  the  B-bar,  is 
proportional  to  H7  or  equal  to  k.^H' ' .  Therefore,  if  T^  =  Tv 
then  k±  —  k2.  The  letter  k  will  be  used  in  what  follows  for  k± 
and  kr 

Consider  the  ship  when  it  is  headed  a  degrees  east  of  north 
as  shown  in  Fig.  16.  The  component  of  Hf  which  magnetizes 
the  A-bar  is  H1  cos  a,  and  the 
magnetic  field  Ta  which  is  pro- 
duced at  C  by  the  magnetiza- 
tion of  the  A-bar  is  k  x  H'  cos 
a.  The  component  of  Hf 
which  magnetizes  the  B-bar  is 
H'  sin  a,  and  the  magnetic  field 
Tb  which  is  produced  at  C  by 
the  magnetization  of  the  B-bar 
is  k  x  H1  sin  a.  The  resultant 


of  Ta  and  Tb  is 


Fig.  16. 


l/T2  +  Th2  =  kH'  V  cos2  a  +  sin2  a  =  kH' 


(0 


Therefore  the  resultant  of  Ta  and  Tb  is  constant  in  value,  and, 
since  Ta  =  kH1  x  cos  a  and  Tb  =  kH1  x  sin  a,  it  is  evident 
that  the  resultant  of  Ta  and  Tb  is  always  opposite  to  Hr  in 
direction,  so  that  the  actual  field  at  the  compass  box  is  constant 
in  value  and  always  parallel  to  H1 ,  or,  in  other  words,  the 
compass  error  due  to  the  temporary  magnetism  of  the  ship  is 
zero  on  all  headings  of  the  ship  when  7^  in  Fig.  14  is  equal  to 
T2  in  Fig.  1 5. 

The  quadrantal  correctors.  —  The  quadrantal  error  of  the  ship's 
compass  is  eliminated  (that  is  to  say,  compensated)  by  means  of 

*  Because  the  magnetization  of  the  A-bar  is  proportional  to   H',    and  the  field   T 
is  proportional  to  the  magnetization  of  the  A-bar. 
21 


306        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

two  iron  spheres  55  which  are  usually  *  placed  on  the  two 
sides  of  the  compass,  as  shown  to  an  exaggerated  scale  in  Fig. 
17.  These  spheres  are  called  tjie  quadrantal  correctors  and  the 

practical  method  of  adjusting 
them  is  explained  in  Art.  16. 
The  action  ..of  the  quadrantal 
correctors  may  be  understood 
with  the  help  of  Figs.  i8#and 
1 8$  as  follows  :  When  the  line 
joining  the  centers  of  the  two  spheres  55  is  parallel  to  Hr  as 
shown  in  Fig.  I  Sa,  the  magnetic  field  at  the  point  /  is  more  in- 
tense than  H1 ;  and  when  the  line  joining  the  centers  of  the 


H' 


H 


Fig.  17. 


Fig.  18a. 


Fig.  18b. 


spheres  is  at  right  angles  to  Hf  as  shown  in  Fig.  i8£,  the  magnetic 
field  at  p  is  less  intense  than  H' .  Now  the  weakening  of  the 
magnetic  field  at  the  compass  box  by  the  temporary  magnetism  of 
the  ship  when  the  ship  heads  east  or  west  is  usually  greater  than 
the  weakening  of  the  field  at  the  compass  box  by  the  temporary 
magnetism  of  the  ship  when  the  ship  heads  north  or  south.  That 


*  In  some  cases,  namely,  when  the  coefficient   kl    is  greater  than  the  coefficient 
it  is  necessary  to  place  the  quadrantal  correctors  fore  and  aft  of  the  coin  pass  box. 


SHIP'S   MAGNETISM.  307 

is  to  say,  T  is  usually  greater  in  Fig.  1 3  than  it  is  in  Fig.  1 2,  or  the 
coefficient  kz  is  usually  greater  than  the  coefficient  kr  There- 
fore by  placing  the  quadrantal  correctors  in  the  positions  shown 
in  Fig.  17  and  moving  them  closer  to  or  farther  away  from  the 
compass,  the  weakening  of  the  magnetic  field  at  the  compass  by 
the  combined  temporary  magnetism  of  ship  and  correctors,  may 
be  made  the  same  with  the  ship's  head  north  (or  south)  as  with 
ship's  head  east  (or  west),  and  when  this  condition  is  reached  the 
quadrantal  error  of  the  compass  is  eliminated  as  explained  above. 

13.  Compass  error  due  to  the  magnetizing  action  of  the  vertical 
component  of  the  earth's  magnetic  field.  —  The  vertical  compo- 
nent V  of  the  earth's  field  produces  a  temporary  magnetism 
of  all  the  vertical  iron  in  the  ship;  this  "  temporary"  magnetism 
remains  unaltered  as  long  as  V  remains  unchanged,  the  ship 
being  supposed  to  stand  on  even  keel ;  and  therefore  the  "  tem- 
porary" magnetism  due  to  V  merges  with  the  permanent  mag- 
netism of  the  ship  in  the  production  of  the  semicircular  compass 
error. 

The  temporary  magnetism  due  to  V  is  distinguishable  from 
the  permanent  magnetism  of  the  ship,  however,  because  it  changes 
when  the  ship  goes  from  one  port  to  another  where  the  value  of 
V  is  different.  Thus,  if  the  semicircular  error  is  completely  com- 
pensated at  the  home  port  by  means  of  the  semicircular  correctors 
(permanent  magnets),  then  a  perceptible  amount  of  semicircular 
error  will  appear  when  the  ship  sails  to  a  distant  port  where  the 
value  of  V  is  different.  In  order  to  overcome  this  difficulty, 
that  is,  in  order  to  compensate  the  semicircular  error  so  that  the 
compensation  may  hold  good  on  a  long  cruise,  it  is  necessary  to 
compensate,  by  means  of  the  semicircular  correctors,  only  that 
part  of  the  semicircular  error  which  is  due  to  permanent  magnetism, 
the  remainder  of  the  semicircular  error  (which  is  due  to  vertical 
temporary  magnetism)  being  compensated  by  means  of  a  vertical 
soft  iron  rod  properly  placed  near  the  compass  box.  The  use  of 
this  rod  was  proposed  originally  by  Captain  Flinders  and  it  is 
usually  called  Flinders'  bar.  The  action  of  Flinders'  bar  may 


308 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


be  explained  as  follows  :  When  V  changes  in  value  the  mag- 
netism of  Flinders'  bar  and  the  vertical  temporary  magnetism  of 
the  ship  change  together,  and  Flinders'  bar  being  once  for  all 
adjusted  to  compensate  the  effect  of  the  vertical  temporary  mag- 
netism of  the  ship,  the  compensation  holds,  whatever  the  value  of  V 
may  be.  Flinders'  bar  is  usually  about  three  inches  in  diameter 
and  from  6  to  24  inches  long,  according  to  the  amount  of  iron 
in  the  vessel,  and  it  is  usually  *  placed  forward  or  aft  of  the 
binnacle. 

*  Figure  iga  shows  the  north  polarity  NNNN,  etc.,  on  the  deck  of  an  iron  vessel 
due  to  the  vertical  component  of  the  earth's  field.  This  north  polarity  is  distributed 
symmetrically  with  respect  to  the  compass  box  C  (ship's  iron  being  symmetrical  with 


bar 


S 


J\ 

I; 


Fig.  19a. 


Fig.  19b. 


respect  to  the  compass  box),  and  it  produces,  at  the  compass,  a  magnetic  field  of  which 
the  horizontal  component  is  represented  by  the  arrow  a  which  is  parallel  to  the  keel. 
Flinders'  bar  is  placed  in  the  position  shown,  and  its  north  pole  Nf  (upper  end 
of  bar),  which  is  on  a  level  with  the  compass  box,  produces  at  the  compass  box  a 
field  b  which  is  equal  and  opposite  to  a.  Figure  iqb  shows  a  side  view  of  Flinders' 
bar  F  (the  compass  box  is  supposed  to  be  placed  at  the  point  p}.  Flinders'  bar 
is  magnetized  by  the  vertical  component  V  of  the  earth's  magnetic  field,  a  is  the 
horizontal  part  of  the  field  which  is  produced  at  the  compass  box  by  the  vertical 


SHIP'S    MAGNETISM.  309 

The  method  of  adjusting  Flinders'  bar  is  explained  in  Art.  1 6. 

14.  The  heeling  error.  —  Let  us  suppose  that  the  semicircular 
and  quadrantal  errors  have  been  completely  compensated  by 
means  of  the  semicircular  correctors  and  quadrantal  correctors, 
the  ship  being  all  the  time  on  an  even  keel.  Under  these  con- 
ditions a  deflection  of  the  compass  is  produced  when  the  ship 
rolls,  or  heels  over,  at  sea.  This  deflection  of  the  compass  is 
called  the  heeling  error,  and  it  is  due  in  part  to  the  variation  of  the 
temporary  magnetism  of  the  ship  which  accompanies  the  change 
of  direction  of  the  earth's  magnetic  field  with  refe/ence  to  the  ship's 
iron  as  the  ship  rolls,  and  in  part  to  the  permanent  magnetism  of 
the  ship,  as  follows :  The  horizontal  field  P,  Fig.  9,  is  annulled 
by  the  semicircular  correctors,  and  the  vertical  component  of  the 
field  at  the  compass  box  which  is  produced  by  the  permanent 
magnetism  of  the  ship  is  left  unaltered  by  the  semicircular  cor- 
rectors. By  vertical  component  is  here  meant  that  component 
which  is  perpendicular  to  the  ship's  deck,  and  which,  as  the  ship 
rolls,  turns  out  of  the  true  vertical,  and  has  a  horizontal  compo- 
nent, at  the  compass,  which  deflects  the  compass.  In  describing 
the  action  of  the  heeling  corrector,  the  heeling  error  will  be  assumed 
to  be  due  entirely  to  the  permanent  magnetism  of  the  ship. 

Assuming  the  heeling  error  to  be  due  entirely  to  the  permanent 
magnetism  of  the  ship,  that  is,  to  be  due  to  the  component  P 
(perpendicular  to  the  deck)  of  the  field  which  is  produced  at  the 
compass  box  by  the  permanent  magnetism  of  the  ship,  it  is  evi- 
dent that  the  heeling  error  is  a  maximum  when  the  ship  heads 
north  or  south,  and  zero  when  the  ship  heads  east  or  west ;  for, 
when  the  ship  heels  over  with  its  head  to  the  east  or  west,  the 
part  of  P'  which  is  projected  upon  a  horizontal  plane  is  directed 
towards  the  north  or  south  and  does  not  deflect  the  compass, 
whereas,  when  the  ship  heels  over  with  its  head  north  or  south, 
the  part  of  Pf  which  comes  into  a  horizontal  plane  is  directed 
towards  the  east  or  west  and  it  deflects  the  compass. 

temporary  magnetism  of  the  ship's  iron,  and   b    is  the  field  produced  at  the  compass 
box  by  Flinders'  bar. 


310        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  heeling  corrector  is  a  vertical  steel  magnet  placed  directly 
beneath  the  compass  box  and  adjusted  up  or  down  until  it  pro- 
duces at  the  compass. box  a  vertical  field  which  is  equal  and  oppo- 
site to  P .  The  practical  method  of  adjusting  the  heeling  cor- 
rector is  explained  in  Art.  16. 

15.  Compass   errors   due   to   magnetic  lag'.- — The   temporary 
magnetism  of  the  ship's  iron   tends  to  lag  behind  the  magnetic 
field  which  produces  it.     Thus,  after  a  ship  has  been  standing  for 
some  time  in  one  direction  the  magnetism  which  is  produced  by 
the  earth's  field  does  not  at  once  disappear  when  the  ship  turns 
around,  but  tends   to   persist.     This   magnetic   lag  produces   a 
compass  error  which  is  known  as  Gaussin's  error  and  which  cannot 
be  compensated. 

16.  Directions  for  adjusting  the  correctors  of  a  ship's  compass. 
(a)  Adjustment  of  semicircular  correctors.  —  The  quadrantal  error 
is  zero  with  ship's  head  north,  east,  south,  or  west.     Therefore 
any  deviation  of  the  compass  which  exists  on  these  headings  is 
due  to  the  semicircular  error.     With  the  ship's  head  north  (mag- 
netic), place  one  or  more  athwartship  magnets  in  one  of  the  semi- 
circular-corrector trays  and   move  them  up  or  down  until  the 
compass  points  north.     Then  head  the  ship  east  (magnetic)  and 
place  fore  and  aft  magnets  in  the  other  semicircular-corrector  tray 
and  move  them  up  or  down  until  the  compass  points  north. 

(b)  Adjustment  of  quadrantal  correctors.  —  Having   corrected 
the  semicircular  deviation  of  the  compass,  head  the  vessel  north- 
east (magnetic)  or  southeast,  southwest,  or  northwest,  and  if  any 
deviation  of  the  compass  exists,  place  the  quadrantal  spheres  on 
the  side  brackets  of  the  binnacle  and  move  them  in  or  out  until 
the  compass  reading  is  correct. 

(c)  Adjustment  of  the  heeling  corrector.  —  With  the  ship  headed 
north  or  south  in  a  heavy  sea,  place  the  heeling-corrector  magnet 
in  its  tube  with  its  proper  end  upwards,  and  raise  or  lower  it 
until  the  slow  motion  of  the  compass  due  to  the  rolling  motion 
of  the  ship  is  nearly  eliminated.     The  proper  end  up  of  the  heeling- 


SHIP'S    MAGNETISM.  311 

corrector  magnet  may  be  inferred  as  follows  :  Suppose  that  the 
north  end  of  the  compass  is  deflected  to  the  east  when  the  ship 
rolls  to  the  west.  Then  it  is  evident  that  the  perpendicular-to- 
the-deck  component  P'  of  the  field  which  is  produced  at  the 
compass  box  by  the  permanent  magnetism  of  the  ship  is  down- 
wards, because  the  part  of  it  which  is  projected  into  a  horizontal 
plane  is  to  the  east  when  the  ship's  masts  roll  to  the  west.  In 
this  case  the  north  end  of  the  heeling- corrector  magnet  is  to  be 
placed  upwards  so  as  to  produce  an  upward  field  at  the  compass 
box. 

(d)  Adjustment  of  Flinders'  bar.  —  Having  carefully  adjusted 
the  semicircular  correctors  at  the  home  port  so  as  to  annul  com- 
pletely the  semicircular  error,  the  ship  is  taken  to  a  distant  port 
and  the  semicircular  error  is  observed  with  the  ship's  head  east 
or  west  (magnetic).  Let  this  error  be  represented  by  <f> ;  let 
V  and  H'  be  the  vertical  and  horizontal  components  of  the 
earth's  magnetic  field  at  the  home  port  and  let  Vl  and  Hf  be 
the  vertical  and  horizontal  components  of  the  earth's  field  at  the 
distant  port  as  determined  by  observation,  or  as  taken  from  mag- 
netic charts.  The  forward  (or  aft)  component  of  the  magnetic 
field  which  is  produced  at  the  compass  box  by  the  vertical  tem- 
porary magnetism  of  the  ship,  is  proportional  to  the  vertical  com- 
ponent of  the  earth's  field  and  it  is  therefore  equal  to  aV  at  the 
home  port  and  equal  to  a  V^  at  the  distant  port.  The  deviation 
of  the  compass  which  is  produced  by  this  field  is  proportional  to 
its  intensity  and  inversely  proportional  to  the  horizontal  intensity 
of  the  earth's  field.  Therefore  this  deviation  is  equal  to  bVf Hr 
at  the  home  port  and  equal  to  b  VJHJ  at  the  distant  port,  where 
a  and  b  are  proportionality  factors.  Therefore  the  observed 
compass  deviation  <£  is  equal  to  b(VjHf—  V^Hf\  and  the 
total  compass  deviation,  <f>t,  which  is  due  to  the  vertical  tem- 
porary magnetism  of  the  ship  at  the  distant  port  is  equal  to 

V  IHf 
rr/rr/      rV  /  TT  f  X  <£.     With  the  ship's  head  east  at  the  distant 

r//T  l/l/fil 

port  (the  condition  under  which    </>    was  observed),  put  Flinders' 


N'J 


312         ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

bar  into  a  vertical  position  in  front  of,  or  behind  the  compass  box, 
and  move  it  towards  or  away  from  the  compass  until  the  compass 
is  turned  through  an  angle  <j)t  in  a  direction  opposite  to  $,  the 

angle    <f>     being   reckoned  from   the  deflected 
NORTJf 

-^\     ^     position   of  the  compass.     Then  eliminate  the 

outstanding  semicircular  error  by    readjusting 
•vjxT-x  \     the  semicircular  correctors. 

Mk 

17.  Napier's  diagram. — After  the  compass 
correctors  have  been  adjusted  so  as  to  approxi- 
mately compensate  the  errors  of  the  compass, 
it  is  customary,  for  the  purpose  of  accurate 
navigation,  to  determine  the  residual  errors  of 
the  compass  and  allow  for  them  in  the  use  of 
the  compass.  The  ship  is  swung  round  and 
for  successive  actual  readings  of  the  compass, 
the  compass  error  is  determined  by  an  inde- 
pendent determination  of  the  true  magnetic 
heading  of  the  vessel.*  Figure  20  shows  Na- 
pier's method  for  representing  the  compass  er- 
rors graphically.  The  successive  actual  com- 
pass readings  are  laid  off  along  the  fine  vertical 
line  as  an  axis,  and  the  compass  errors  are  laid 
off  along  the  fine  dotted  lines  which  are  in- 
clined at  an  angle  of  60°  to  the  fine  vertical  line. 
To  determine  the  true  magnetic  course  of 
the  ship  from  the  compass  reading,  start  at 
the  point  on  the  vertical  axis  which  corresponds 
to  the  actual  compass  reading,  draw  a  line  parallel  to  the  fine 
dotted  lines  from  the  chosen  point  on  the  vertical  axis  to  the 
curve  of  errors  (which  is  the  heavy  curve  in  Fig.  20)  ;  from  the 
point  so  reached  on  the  curve  of  errors,  draw  a  line  parallel  to 

*  The  true  magnetic  heading  is  determined  by  a  land-mark,  if  the  vessel  is  in  port, 
or  by  observations  on  the  sun  or  stars  if  the  vessel  is  at  sea,  the  declination  of  the 
compass  being  known  for  the  place  of  observation. 


SHIP'S   MAGNETISM.  313 

the  fine  full  lines,  and  the  point  where  this  line  cuts  the  vertical 
axis  corresponds  to  the  true  magnetic  course  of  the  vessel.  To 
determine  the  compass  reading  corresponding  to  a  true  magnetic 
course,  start  from  a  point  on  the  vertical  axis  which  corresponds 
to  the  true  magnetic  course,  travel  parallel  to  the  fine  full  lines 
until  the  curve  of  errors  is  reached  and  then  travel  parallel  to  the 
fine  dotted  lines  until  the  desired  point  on  the  vertical  axis  (corre- 
sponding to  the  actual  compass  reading)  is  reached. 

The  heavy  curve  in  Fig.  20  represents  the  actual  compass 
errors  on  the  old  British  iron-clad  Achilles,  and  the  abscissas  of 
the  fine  sine  curves  represent  the  semicircular  errors  and  quad- 
rantal  errors,  respectively.  The  maximum  value  of  the  semicir- 
cular error  is  21°  15',  and  the  maximum  value  of  the  quadrantal 
error  is  6°  9'. 

PROBLEMS. 

1.  The  semicircular  error  of  a  compass  on  board  ship  is  found 
to  have  a  maximum  value  of  20°  to  the  east  when  the  ship  heads 
36°  west  of  south.     Make  a  sketch  of  the  outline  of  the  deck 
of  the  vessel  and  draw  a  line  on  the  deck  showing  the  direction 
of  the  horizontal  component  of  the  magnetic  field  at  the  compass 
box  which  is  due  to  the  permanent  magnetism  of  the  ship,  find  the 
value  of  this  horizontal  component  and  find  the  angle  between  its 
direction  and  the  direction  of  the  keel,  the  earth's  horizontal  field 
being  equal  to  0.2  gauss.     Ans.  (a)  0.068  gauss,  (^)  106°  from 
bow  towards  port  side  (left  side). 

2.  What  is  the  value  of  the  semicircular  error  of  the  compass 
when  the  ship  specified  in  problem    I    heads  20°  north  of  east  ? 
Ans.  19°  21',  west  of  north. 

3.  The  only  error  of  a  ship's  compass  is  that  which  is  due  to 
the  ship's  permanent  magnetism,  the  quadrantal  error  being  com- 
pensated.    The  semicircular  error  has  a  value  of  6°  to  the  west 
when' the  ship's  head  is  true  magnetic  north  and  4°  to  the  west 
when  the  ship's  head  is  true  magnetic  northeast.      On  what  true 
headings  will  the  error  of  the  compass  be  zero  ?     Ans.  38  minutes 
south  of  east,  and  38  minutes  north  of  west. 


314        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

4.  Suppose  that  the  semicircular  error  of  the  ship's  compass 
has   been  completely  compensated  and  suppose  that  the  quad- 
rantal  error  is  observed  to  be  4°  to  the  west  when  the  ship  is  headed 
true  northeast.     What  is  the  cleviation  of  the  compass  when  the 
ship  heads  30°  south  of  east.     Ans.  3°  20'  to  the  east. 

Note.  —  In  this  problem  treat  the  ship  as  one  long  slim  iron  bar  parallel  to  the 
keel. 

5.  (a)  A  ship  is  headed  true  magnetic  north  (for  which  posi- 
tion the  quadrantal  error  is  zero),  and  the  compass  shows  a  devia- 
tion to  the  east.     A  permanent  magnet  is  to  be  placed  in  an  east- 
west  direction  (athwartship)  underneath  the  compass  box  so  as 
to  bring  the  compass  to  true  magnetic  north.     Which  end  of  the 
magnet  is  to  be  placed  to  the  east  ?     (ft)  The  ship  is  then  headed 
true  magnetic  east  and  the  compass  is  observed  to  have  a  devia- 
tion to  the  west.     A  permanent  magnet  is  to  be  placed  in  an  east- 
west  direction  (parallel  to  the  keel)  underneath  the  compass  box 
so  as  to  bring  the  compass  to  true  magnetic  north.     Which  end 
of  the  magnet  is  to  be  placed  to  the  east  ?     Ans.  (a)  north  end 
east,  (&)  north  end  west. 

6.  The  semicircular  and  quadrantal  errors  having  been  com- 
pensated the  ship  is  headed  magnetic  south  at  sea  and  the  com- 
pass is  deflected  to  the  west  when  the  ship  heels  to  the  east  (top 
of  mast  moves  eastward).     Which  end  of  the  heeling  corrector 
magnet  must  be  placed  upwards  in  order  to  eliminate  the  heeling 
error  ?     Ans.  North  end  up. 


APPENDIX  C. 
MISCELLANEOUS  PHENOMENA. 

18.  Thermo-electricity.*  Seebeck's  discovery.  —  In  1821  See- 
beck  found  that  an  electric  current  is  produced  in  a  circuit  of  two 
metals  when  one  of  the  junctions  of  the  two  metals  is  warmer 
than  the  other.  Seebeck  used  the  arrangement  shown  in  Fig.  2 1 . 

The  ends  of  a  bent  bar  of  copper 
were  soldered  to  the  ends  of  a 
rod  of  bismuth,  a  magnetic  needle 
was  pivoted  between  the  bars  as 
shown  in  the  figure,  and  one  of 

Fig.  21. 

the  junctions  was  heated  by  a 

spirit  lamp.  The  existence  of  current  is  indicated  by  the  de- 
flection of  the  magnetic  needle,  and  the  direction  of  the  current 
which  is  produced  is  shown  by  the  arrows  in  Fig.  21.  An 
arrangement  such  as  is  shown  in  Fig.  2 1  is  called  a  thermo-element. 

The  thermopile.  —  The  electromotive  force        1357 
of  a  single  thermo-element  seldom  exceeds  a    A/V/VX  \X  \ 
few  thousandths  of  a  volt,  even  when  the  two 
junctions  are  at  widely  different  temperatures. 
A  number  of  thermo-elements  may,  however, 
be  connected  in  series,  as  in  Fig.  22,  in  which 
AAAA  are  bars  of  one  metal  and  BBBB  are 
bars  of  another  metal.     Junctions  I,  3,  5  and 
7  are  heated,  while  junctions  2,  4  and  6  are  kept  cool,  or  vice 
versa. 

The  thermo-element  used  as  a  pyrometer. \  —  When  one  junction 
of  a  thermo-element  is  kept  at  a  constant  standard  temperature, 

*  A  very  good  discussion  of  Thermo-electricity  is  given  in   Magnetism  and  Elec- 
tricity for  StudentsbyH.  E.  Hadley,  pages  359-382,  Macmillan  and  Company,  1906. 
f  A  pyrometer  is  a  thermometer  for  measuring  very  high  temperatures. 

315 


316        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

the  electromotive  force  of  the  element  is  a  function  of  the  tem- 
perature T  of  the  other  junction,  and,  if  the  electromotive  force 
of  the  element  is  determined  once  for  all  for  a  series  of  values  of 
Tt  then  any  unknown  temperature  may  be  determined  by  observ- 
ing the  electromotive  force  of  the  thermo-element  when  one  of 
its  junctions  is  at  the  standard  temperature  and  the  other  is  at  the 
temperature  which  is  to  be  measured. 

The  electromotive  force  of  a  thermo-element  can  *  be  repre- 
sented with  a  fair  degree  of  accuracy  by  the  equation 

e  =  a  +  bT+cT2  (i) 

when  one  junction  of  the  element  is  kept  at  a  fixed  standard  tem- 
perature, a,  b  and  c  being  constants.  Therefore  in  order  to 
use  a  thermo-element  as  a  pyrometer,  it  is  sufficient  to  measure  the 
electromotive  force  e  of  the  element  for  three  chosen  known 
values  of  T.  The  thermo-element  which  has  proved  most  satis- 
factory for  use  as  a  pyrometer  is  one  employing  pure  platinum 
and  an  alloy  of  platinum  and  rhodium. 

The  Peltier  effect.  —  In  1834  Peltier  discovered  that  heat  (in- 
dependently of  the  heat  generated  in  accordance  with  Joule's 
Law,  Art.  12,  Chapter  II)  is  generated  or  absorbed  at  a  junc- 
tion of  two  metals  when  a  current  flows  across  the  junction,  that  is, 
heat  is  generated  when  the  current  flows  in  one  direction  and 
absorbed  when  the  direction  of  the  current  is  reversed  ;  the  gen- 
eration of  heat  being  shown  by  an  increase  of  temperature  of  the 
junction,  and  the  absorption  of  heat  being  shown  by  a  cooling  of 
the  junction.  For  strong  currents  this  Peltier  effect  is  masked 
by  the  heat  that  is  generated  on  account  of  electrical  resistance, 
for  the  rate  of  generation  of  heat  by  the  Peltier  effect  is  proportional 
to  the  current,  while  the  rate  of  generation  of  heat  on  account  of 
resistance  is  proportional  to  the  square  of  the  current.  The 
Peltier  effect  is  most  easily  shown  as  follows  :  A  current  from  a 
voltaic  cell  is  sent  through  a  thermopile.  This  current  heats  one 
set  of  junctions  and  cools  the  other  set.  The  thermopile  is  then 

*See  Magnetism  and  Electricity  for  Students,  H.  E.  Hadley,    pages  361-367. 


MISCELLANEOUS  PHENOMENA.          4  I/ 

disconnected  from  the  voltaic  cell  and  connected  to  a  galvanometer 
and  the  difference  in  temperature  of  the  two  sets  of  junctions  is 
shown  by  the  deflection  of  the  galvanometer. 

The  Thomson  effect.  —  When  a  liquid,  like  water,  flows  along 
a  pipe  which  is  not  at  a  uniform  temperature,  the  liquid  always 
absorbs  heat  from  the  pipe  at  each  point  when  it  flows  in  the  direc- 
tion of  increasing  temperature  of  pipe,  and  the  liquid  always  gives 
out  heat  to  the  pipe  at  each  point  when  it  flows  in  the  direction  of 
decreasing  temperature.  Lord  Kelvin  (then  Sir  William  Thom- 
son) discovered  in  1851  that  an  electric  current  may  either  absorb 
or  give  out  heat  at  each  point  in  a  wire  when  the  temperature  of 
the  wire  is  not  uniform.  If  the  electric  current  absorbs  heat  at  each 
point  of  a  wire  when  it  flows  along  a  wire  in  the  direction  of  in- 
creasing temperature,  the  Thomson  effect  is  considered  to  be  posi- 
tive. If  the  electric  current  gives  out  heat  at  each  point  when  it 
flows  in  the  direction  of  increasing  temperature,  the  Thomson 
effect  is  considered  to  be  negative. 

19.  Pyro-electricity.* — A  peculiar  property  of  a  crystal  of 
tourmaline  after  its  temperature  had  been  increased  or  decreased 
was  noted  by  Daumius  in  1 707.  The  crystal  had  the  property  of 
attracting  small  particles  of  ashes.  Aepinus  in  1756  recognized 
this  property  of  a  tourmaline  crystal  as  an  electrical  phe- 
nomenon, and  he  was  able  to  show  that  the  two  ends  of  a 
tourmaline  crystal  become  oppositely  charged  when  the  tempera- 
ture of  the  crystal  is  changed.  Very  extensive  experimental 
studies  of  the  production  of  electric  charges  on  the  surface  of 
crystals  by  changes  of  temperature  were  carried  out  by  Hankel, 
beginning  in  1839.  Hankel  found  that  the  property  of  becoming 
charged  by  a  change  of  temperature  is  common  to  all  crystals, 
although  hemihedral  crystal  forms  show  the  effect  more  strik- 
ingly. A  method  for  demonstrating  this  so-called  gyro-electric 
property  of  crystals  is  to  place  a  mixture  of  finely-powdered  sul- 
phur and  red  lead  in  a  fine  cotton  sieve  and  dust  it  upon  the 
crystal  after  the  temperature  of  the  crystal  has  been  changed. 

*  See  Wiedemann,  Die  Lehre  von  der  Elektricitdt,  Vol.  II,  pages  316-340. 


318        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

The  effect  of  the  cotton  sieve  is  to  give  a  negative  charge  to  the 
sulphur  particles  and  a  positive  charge  to  the  red  lead  particles, 
so  that  the  sulphur  particles  cling  to  the  positively  charged  parts 
of  the  crystal  and  the  red  lead  -Jarticles  cling  to  the  negatively 
charged  parts  of  the  crystal. 

Piezo-electricity*  —  In  1880  it  was  found  by  J.  and  P.  Curie 
that  many  kinds  of  crystals  become  electrically  charged  when 
they  are  subjected  to  pressure.  This  effect  is  produced  in  hemi- 
hedral  crystal  forms  when  a  crystal  plate  with  its  faces  at  right 
angles  to  the  hemihedral  axis  is  compressed  between  two  metal 
plates.  The  effect  is  to  charge  the  two  metal  plates  oppositely. 

20.  Magnetic  rotation  of  the  plane  of  polarization  of  light.  — 
Faraday  f  discovered  in   1 846  that  a  transparent  substance  such 
as  glass  or  carbon  bi-sulphide  rotates  the  plane  of  polarization  of 
light  when  it  is -placed  in  the  magnetic  field  and  when  the  light  is 
passed  through  it  in  the  direction  of  the  magnetic  lines  of  force. 

21.  The  Hall  effect.  J  —  When  a  conductor  through  which  an 
electric  current  is  flowing  is  placed  in  a  magnetic  field,  the  con- 
ductor is  acted  upon  by  a  force  which  pushes  it  sidewise  as  ex- 
plained in  Chapter  IV.     Ordinarily  this  force  does  not  alter  the 
distribution  of  current  in  the  conductor,  that  is  to  say,  the  cur- 
rent is  not  pushed  to  one  side  of  the  conductor.     E.  H.  Hall 
discovered  in   1880,  however,  that  the  current  is  pushed  to  one 
side  of  the  conductor  to  a  very  slight  extent  in  some  metals, 
especially  in  bismuth.     This  peculiar  effect  is  satisfactorily  ex- 
plained by  the  electron  theory  of  metallic  conduction  (see  Lodge's 
Electrons,  pages  106-109). 

22.  The  Kerr  effect.  §  —  One  of  the  most  universally  applicable 
principles  in  physics  is  the   principle  of  superposition,  so-called, 

*  See  Wiedemann,  Die  Lehre  von  der  Elektricitat,  Vol.  II,  pages  341-346,  and 
Vol.  IV,  pages  1280-1284. 

f  See  Faraday's  Experimental  Researches,  Series  19,  1846.  A  description  of 
Faraday's  experiments  and  of  later  experiments  along  the  same  line  is  given  in 
Wiedemann,  Die  Lehre  von  der  Elektridtdt,  Vol.  Ill,  pages  907-968. 

J  See  Wiedemann,  Die  Lehre  von  der  Elektricitdt,  Vol.  Ill,  pages  192-194. 

§  See  Wiedemann,  Die  Lehre  von  der  Elektricitat,  Vol.  II,  pages  126-136. 


MISCELLANEOUS   PHENOMENA. 


319 


which  in  its  most  general  form  may  be  stated  as  follows  :  Given 
a  cause  which  produces  an  effect  which  is  proportional  to  it. 
Then  two  such  causes  acting  together  produce  an  effect  which  is 
the  sum  of  the  effects  which  they  would  produce  if  they  acted 
separately,  and  the  totaj.  effect  may  be  divided  into  two  parts 
which  correspond  to  the  two  parts  of  the  cause,  or  in  other 
words,  each  cause  produces  the  same  effect  that  it  would  produce 
if  it  were  acting  by  itself.  One  of  the  best  examples  of  this 
principle  is  that  light  passes  from  two  windows,  for  example, 
through  the  same  region  to  the  eyes  of  two  observers  and  each 
observer  sees  his  window  distinctly,  that  is  to  say,  the  light 
travels  through  the  given  region  from  each  window  exactly  as  if 
it  were  traveling  through  the  region  alone.  This  principle  of 
superposition  is  quite  accurately  true  in  most  of  the  phenomena 
of  the  electromagnetic  field.  It  was  discovered,  however,  by 
Kerr,  in  1875  that  an  isotropic  transparent  substance  such  as 
glass  or  oil  becomes  doubly  refracting  when  subjected  to  a  strong 
electric  field. 

23.  The  Zeeman  effect.*  —  About  1900  it  was  predicted  by 
Lorenz  and  experimentally  verified  by  Zeeman,  that  the  light 
emitted  by  a  hot  vapor  is  altered  in  a  peculiar  way  when  the 
vapor  is  placed  in  an  intense  magnetic  field.  The  character  of 
this  alteration  when  the  emitted  light  travels  parallel  to  the  lines 
offeree  of  the  magnetic  field  is  as  follows  :  Imagine  an  atom  to 
consist  of  a  positively  charged  nucleus  with  one  or  more  nega- 


A  B 

Fig.  23. 
*See  Lodge's  Electrons,  pages  109-115. 


320        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

tively  charged  satellites  revolving  round  the  nucleus  as  repre- 
sented in  Fig.  23.  Imagine  the  region  in  Fig.  23  to  be  a  mag- 
netic field  directed  towards  the  reader.  The  effect  of  such  a  mag- 
netic field  would  be  to  push  outwards  on  the  satellite  a,  thus 
increasing  its  periodic  time  of  revolution,  whereas  the  effect  would 
be  to  push  inwards  on  the  satellite  a' ',  thus  decreasing  its  peri- 
odic time  of  revolution.  Now  the  hypothesis  which  has  been 
used  in  the  discussion  of  the  Zeeman  effect  is  that  a  given  line  of 
the  spectrum  of  the  hot  vapor  is  due  to  the  rotation  of  a  certain 
satellite  in  the  atom,  at  a  certain  speed.  The  plane  of  the  orbit 
of  this  particular  satellite  has  every  possible  orientation  in  the  dif- 
ferent atoms  of  the  vapor  as  shown  by  A,  B  and  C,  Fig.  23, 
and,  when  the  vapor  is  not  in  a  magnetic  field,  the  periodic  time  of 
rotation  of  the  satellite  a  is  the  same  for  all  the  atoms.  When, 
however,  the  vapor  is  in  the  magnetic  field,  the  periodic  time  of 
the  satellite  a  is  increased,  the  periodic  time  of  satellite  a'  is 
decreased,  and  the  periodic  time  of  satellite  a",  the  plane  of 
whose  orbit  is  parallel  to  the  magnetic  field,  is  unaltered.  There- 
fore, instead  of  one  single  spectrum  line  corresponding  to  the  given 
satellite,  there  will  be  three  lines,  one  in  the  original  position  and 
one  on  each  side  of  the  original  position. 

24.  Lippmann's  electrometer.*  —  A  pool  of  mercury  underneath 
an  electrolyte,  such  as  dilute  sulphuric  acid,  can  of  course  be 
used  as  an  electrode  of  an  electrolytic  cell.  When  this  is  done 
the  surface  tension  of  the  mercury  is  altered,  the  change  of  sur- 
face tension  being  approximately  proportional  to  the  polarization 
electromotive  force  (electromotive  force  between  the  metal  and 
the  electrolyte).  This  change  of  surface  tension  of  mercury  by 
electrolytic  polarization  may  be  demonstrated  by  the  change  in 
level  of  a  mercury  column  in  a  capillary  tube  when  the  surface 
of  the  mercury  column  is  polarized.  This  effect  was  discovered 
about  1870  and  it  was  employed  by  Lippmann  in  the  construction 
cf  a  capillary  electrometer  in  which  the  movement  of  a  mercury 

*  See  Wiedemann,  Die  Lehre  von  der  Elektricitat,  Vol.  II,  pages  708-720,  for  a 
full  discussion  of  the  polarization  of  mercury. 


MISCELLANEOUS    PHENOMENA. 


321 


column  in  a  capillary  tube  is  used  as  an  indicator  of  electromotive 
force. 

25.  Electric   osmosis.  *  —  A    U-tube   AB,    Fig.    24,   is    filled 
with  water  and  provided  with  two  platinum  electrodes,  and  an 
electric  current  is  sent  through  the  cell  in  the  direction  of  the 
arrows.     The  bend  of  the  tube  is  filled  with  fine  sand.     Under 
these  conditions  the  water  is 

found  to  rise  in  the  arm  B 

and  fall  in  the  arm    A,  or, 

in  other  words,  the  current 

causes  the  water   to   diffuse 

through  the  sand  from  A  to 

B.     This  forced  diffusion  of 

a   liquid   through   a  porous 

diaphragm  is  called  electric 

osmosis.     It  was  discovered 

in    1807  by    Reuss.       This 

effect    is     greatly     reduced 

when  a  good  conducting  liquid,  such  as  an  acid  or  salt  solution, 

is  used  instead    of  water.      In  1835  Becquerel   discovered  that 

fine  particles  of  clay  or  other  material   suspended  in  water  are 

caused  to  travel  in  one  direction  or  the  other  when  an  electric 

current  is  sent  through  the  water. 

26.  The  change  of  electrical  resistance  of  selenium  by  illumi- 
nation, f — Willoughby  Smith,  in  1873,  discovered  that  the  elec- 
trical resistance  of  metallic  selenium  is  reduced  to  one  half  or  one 
third  of  its  normal  value  when  the  selenium  is  exposed  to  brilliant 
sunlight. 

27.  Atmospheric    electricity.  —  It   was    shown    by    Benjamin 
Franklin  about  1760  that  the  lightning  discharge  is  identical  in 
its  nature  to  the  ordinary  electric  spark.     Very  little  was  learned 
.after  Franklin's  time  concerning   the  cause  of  atmospheric  elec- 

*See  Wiedemann,  Die  Lehre  von  der  Elektricitdt,  Vol.  II,  pages  166-195. 
f  See  Wiedemann,  Die  Lehre  von  der  Elektricitat,  Vol.  I,  pages  547-551. 
22 


322       ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

tricity  until  about  1 896  when  the  electron  theory  had  been  de- 
veloped. The  present  theory  of  atmospheric  electricity  as  devel- 
oped chiefly  by  Wilson,  of  Cambridge,  England,  is  as  follows  : 

4J| 

When  moist  air  is  cooled,  the  wa'ter  vapor  always  becomes  super- 
saturated unless  there  are  nuclei  present  upon  which  the  water 
vapor  can  condense.  It  has  been  experimentally  demonstrated 
that  both  positive  and  negative  ions  can  serve  as  condensation 
nuclei,  and  that  a  lower  degree  of  super-saturation  is  required  to 
cause  the  negative  ions  to  act  as  condensation  nuclei  than  is  re- 
quired to  cause  the  positive  ions  to  act  as  condensation  nuclei. 
The  upper  regions  of  the  atmosphere  where  the  ultra-violet  rays  of 
the  sun's  light  are  very  intense,  are  strongly  ionized,  and,  when  the 
water  vapor  in  these  upper  regions  becomes  super-saturated  by 
cooling,  the  negative  ions,  becoming  loaded  by  the  condensation 
of  moisture,  fall  towards  the  earth  leaving  the  upper  regions  of 
the  atmosphere  positively  electrified.  The  great  intensity  of  the 
electric  phenomena  of  the  atmosphere  during  the  summer  time  is 
probably  due  to  the  fact  that  during  the  summer  the  condensa- 
tion of  moisture  takes  place  at  very  great  altitudes  where  the  ioni- 
zation  of  the  atmosphere  is  very  great,  whereas  during  the  winter 
time  most  of  the  condensation  which  takes  place  occurs  at  very 
much  lower  altitudes  where  the  atmosphere  is  not  strongly  ionized. 
Lightning  protection.  —  The  use  of  the  lightning  arrester 
for  protecting  electrical  machinery  is  described  in  Chapter  VI. 
The  use  of  the  lightning  rod  for  the  protection  of  buildings 
against  damage  by  lightning  is  due  to  Benjamin  Franklin.  A 
lightning  rod  is  simply  a  good  conductor  leading  as  directly  as 
possible  from  a  point  above  a  building  to  a  good  ground  connec- 
tion in  moist  earth.  A  house  which  is  not  guarded  by  a  lightning 
rod  may  not  be  damaged,  and,  in  many  cases,  houses  which  are 
guarded,  are  severely  damaged,  but  statistics  show  that  the  num- 
ber of  casualties  is  very  greatly  reduced  by  the  use  of  lightning 
rods.  There  is  therefore  no  question  as  to  the  usefulness  of  the 
lightning  rod.  Information  concerning  lightning  rods  may  be 
obtained  from  Sir  Oliver  Lodge's  book  Lightning  Conductors  and 
Lightning-  Guards,  Whitaker  &  Co.,  London,  1892. 


APPENDIX  D. 
MISCELLANEOUS  PRACTICAL  APPLICATIONS.* 

28.  The  Morse  telegraph  is  an  arrangement  for  signalling 
between  distant  stations  as  follows  :  An  insulated  wire  leads  from 
one  station  to  the  other  and  back.  The  ground  is  generally  used 
instead  of  a  return  wire.  An  electric  current  from  a  battery  or 
other  source  is  sent  intermittently  through  this  circuit  by  operat- 
ing at  one  station  a  key  which  makes  and  breaks  the  circuit. 
This  current  excites  an  electromagnet  at  the  other  station,  and 
the  armature  of  this  electromagnet  makes  a  graphical  record  on 
a  moving  strip  of  paper,  or  produces  sound  signals  which  are 
interpreted  by  the  operator  at  the  receiving  station. 

Relays  and  sounders.  —  A  fairly  strong  electric  current  is 
required  to  operate  the  instrument  which  produces  the  signals  at 
a  telegraph  receiving  station,  and  it  is  not  desirable  to  send  so 
strong  a  current  over  a  long  line  because  of  the  great  number  of 
voltaic  cells  that  would  be  required.  This  difficulty  is  obviated 
by  the  use  of  the  relay.  The  current  in  the  line  flows  through 


Fig.  25- 

many  turns  of  fine  wire  which  are  wound  upon  an  electromagnet 
at  the  receiving  station.  This  magnet  actuates  a  very  light  lever 
and  this  lever  is  arranged  to  open  and  close  what  is  called  a  local 

*  Many  of  the  practical  applications  of  electricity  and  magnetism  have  been  described 
in  the  foregoing  chapters. 

323 


324        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

circuit  as  it  moves  back  and  forth  between  stops.  Figure  25  is  a 
view  of  such  an  instrument,  which  is  called  a  relay.  The  local 
circuit  which  is  opened  and  closed  by  the  relay  contains  a  battery 
which  supplies  the  large  current  that  is  required  for  the  operation 
of  the  instrument  which  produces  the  sound  signals.  This 
instrument  is  called  a  sounder.  It  consists  of.  an  electromagnet, 


Fig.  26.  Fig.  27. 

which  is  wound  with  moderately  coarse  wire  and  which  actuates 
a  massive  lever  and  produces  audible  signals  as  it  moves  back 
and  forth  between  stops.  Figure  26  shows  the  ordinary  tele- 
graph sounder.  Figure  27  shows  an  ordinary  telegraph  key. 

29.  The  polarized  relay.  —  The  ordinary  relay  which  is  shown 
in  Fig.  25  responds  to  a  make-and-break  key.  By  using  the 
proper  tension  on  the  spring  which  pulls  the  lever  back  (see  Fig. 
25),  the  lever  of  the  ordinary  relay  may  be  made  to  respond  to 
increase  and  decrease  of  current,  whereas  a  quick  reversal  of  cur- 
rent may  not  affect  the  instrument,  inasmuch  as  the  lever  may  not 
have  time  to  move  perceptibly  while  the  current  is  passing 
through  zero  value. 

The  polarized  relay  is  so  constructed  as  to  respond  to  reversals 
of  current,  but  not  to  respond  to  increase  and  decrease  of  current. 
An  electromagnet  NNV  Fig.  2%a,  is  mounted,  as  shown,  upon 
one  pole  of  a  U-shaped  permanent  magnet.  A  light  iron  lever 
a,  Fig.  28(5,  pivoted  at  /,  passes  through  a  slot  in  the  south 
pole  55  of  the  permanent  magnet,  between  the  poles  NN^  of 
the  electromagnet,  and  plays  between  the  stops  /'  and  pn '. 
This  lever  a  is  magnetized  inasmuch  as  it  bridges  over  from  the 


MISCELLANEOUS    PRACTICAL   APPLICATIONS. 


325 


south  pole  55"  of  the  permanent  magnet  to  the  soft  iron  cores 
NN^  which  stand  upon  the  north  pole  of  the  permanent  magnet. 
When  a  current  flows  in  a  certain  di- 
rection through  the  coils  of  the  elec- 
tromagnet N^  one  of  its  poles,  Nv 
for  example,  becomes  a  strong  north 
pole  and  attracts  the  lever  a.  When 
the  current  is  reversed,  the  other  pole 
N  becomes  a  strong  north  pole  and 
attracts  the  lever  a.  Thus,  the  lever 
a  is  pulled  towards  N^  or  towards  N 
according  to  the  direction  of  the  cur- 
rent which  flows  through  the  coils  of 
the  instrument,  and  a  local  circuit  con- 
nected, as  shown  in  Fig.  28$,  may  thus  be  opened  and  closed  at 
will  by  repeated  reversals  of  the  current  through  the  winding 
of  the  electromagnet  NNr 

The  ordinary  relay  is  usually  called  the  neutral  relay  to  dis- 
tinguish it  from  the  polarized  relay. 


N -  POLE 
STEEL  MAGNET 


Fig.  28a. 


w  . 

0= 

JO  life 

Qs-' 

LOCAL 
CIRCUIT 


LOCAL   y 
CIRCUIT' 


Fig.  28b. 


30.  Diplex  telegraphy.  —  The  sending  of  two  messages  in  the 
same  direction  over  one  line  wire  simultaneously  is  known  as 
diplex  telegraphy.  This  is  accomplished  as  follows :  At  the  send- 
ing station  are  two  keys.  One  of  these  keys  is  arranged  to  vary 
the  strength  of  the  current  in  the  line  (never  actually  breaking 
the  circuit)  by  throwing  a  number  of  voltaic  cells  in  and  out  of 
circuit  as  it  is  operated.  The  other  key  is  arranged  to  reverse 
the  direction  of  the  line  current  as  it  is  operated,  the  line  current 
being  in  one  direction  while  this  key  is  down,  and  in  the  other 


326        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

direction  while  it  is  up.  At  the  receiving  station  a  neutral  relay 
and  a  polarized  relay  are  connected  in  circuit  with  the  line.  The 
neutral  relay  responds  to  the  l£ey  which  varies  the  strength  of 
the  line  current,  and  the  polarized  relay  responds  to  the  key 
which  reverses  the  line  current. 

31.  Duplex  telegraphy. —  The  sending  of  two"  messages  in  oppo- 
site directions   over  one   line  wire   simultaneously  is   known   as 
duplex  telegraphy.     This  is  accomplished  as  follows  :  Fig.  29  rep- 
resents the  arrangement  of  apparatus  at  one  station.     An  exactly 

similar  arrangement  is 
installed  at  the  other 
station.  Let  c  be  the 
total  resistance  of  the 
line  through  the  distant 
station  to  the  ground. 
Then  the  resistances  a, 

&,  c  and  d  form  a  Wheat- 
Fig.  29.  ,       ,     .  ,  TT71 

stone  s    bridge.      When 

these  resistances  are  so  adjusted  that  ajb  =  cjd,  then  the  key 
at  the  home  station  may  be  pressed  without  sending  a  current 
through  the  home  relay.  When  the  key  at  the  home  station  is 
pressed,  however,  current  flows  over  the  line  to  the  other  station, 
and  it  is  easily  seen  from  the  figure  that  a  line  current  coming  to 
a  station  divides,  and  flows  in  part  through  the  relay  at  that  sta- 
tion. Therefore  the  relay  at  each  station  responds  to  the  move- 
ments of  the  key  at  the  other  station. 

32.  Quadruplex  telegraphy.  —  The  sending  of  two  messages 
each  way  over  one  line  wire  simultaneously  is  known  as  quadru- 
ple* telegraphy.    This  is  accomplished  by  combining  the  arrange- 
ments for  diplex  and  duplex  telegraphy.     The  single  key  repre- 
sented in  Fig.  29  is  replaced  by  two  keys,  one  for  reversing  the 
current  and  the  other  for  altering  its  strength  ;  and  the  single 
relay  is  replaced  by  two  relays,  one  a  neutral  relay  and  the  other 
a  polarized  relay.     With  this  arrangement  the  polarized  relay  at 


MISCELLANEOUS   PRACTICAL  APPLICATIONS.          327 

each  station  responds  to  the  reversing  key  at  the  other  station, 
and  the  neutral  relay  at  each  station  responds  to  the  key  at  the 
other  station  which  alters  the  strength  of  the  current. 

33.  The  printing  telegraph  is  an  arrangement  by  means  of  which 
a  simple  form  of  typewriter  is  operated  at  a  distant  station  from  a 
keyboard  at  a  sending  station.  A  simple  form  of  printing  tele- 
graph is  as  follows  :  *  Twenty -six  equidistant  pins  are  arranged 
in  a  helical  row  around  a  long  metal  cylinder.  This  cylinder  is 
rotated  by  a  small  electric  motor  or  by  clockwork,  and  above  the 
cylinder  is  a  bank  of  twenty-six  lettered  keys  so  arranged  that 
when  a  key  is  depressed,  one  of  the  pins  comes  against  it  and  the 
cylinder  is  stopped  in  a  certain  position ;  the  next  key  would 
stop  the  cylinder  ^  of  a  revolution  farther  on,  and  so  on.  At- 
tached to  the  rotating  cylinder  is  a  device  for  reversing  an  elec- 
tric current  fifty-two  times  for  each  revolution  of  the  cylinder. 
This  repeatedly  reversed  electric  current  passes  over  the  tele- 
graph line  and  through  two  electromagnets  at  the  receiving 
station.  One  of  these  electromagnets  is  like  a  neutral  relay  with 
a  heavy  lever,  and  the  other  is  like  a  polarized  relay  with  a  light 
lever  which  oscillates  with  the  rapid  reversals  of  current  and 
actuates  an  escapement  which  turns  a  type  wheel  with  the  twenty- 
six  letters  arranged  round  its  periphery.  This  type  wheel  is  thus 
turned  step  by  step,  keeping  pace  with  the  rotating  cylinder  at 
the  sending  station. 

When  the  cylinder  at  the  sending  station  is  stopped  by  de- 
pressing a  key,  the  A-key,  for  example,  the  current-reversing 
device  stops  also,  a  steady  current  flows  over  the  line,  the  tongue 
of  the  polarized  relay  stops  oscillating,  the  type  wheel  stops,  and 

*  When  a  person  is  thoroughly  familiar  with  the  elements  which  enter  into  the  con- 
struction of  a  machine,  that  is,  when  a  person  is  familiar  with  shafts  and  wheels,  and 
with  simple  devices  like  switches  for  opening  and  closing  electric  circuits  and  for  re- 
versing connections,  a  more  easily  intelligible  description  of  a  complicated  machine 
can  be  made  without  illustrative  diagrams  and  drawings  than  can  be  made  with  the 
help  of  diagrams  and  drawings.  In  fact,  it  is  confusing  under  the  specified  conditions 
to  have  recourse,  even,  to  a  working  model  of  a  complicated  machine,  when  the  object 
in  view  is  to  impart  a  clear  idea  of  its  fundamental  features. 


328 


ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


letter  A. 


the  steady  current  excites  the  neutral  relay,  the  lever  of  which 
pushes  a  strip  of  paper  against  the  type  wheel  and  prints  the 
When  the  key  at  the  sending  station  is  raised,  the  cur- 
rent reversals  begin  again,  the  type 
wheel  at  the  receiving  station  starts, 
and  at  the  same  time  the  lever  of  the 
neutral  relay  falls  back  and  actuates  a 
device  which  moves  the  strip  of  paper  a 
step  forward  for  the  printing  of  the 
next  letter. 


Fig.  30. 


34,  Submarine  telegraphy.  —  Figure 
30  shows  a  full-size  sectional  view  of  a 
submarine  telegraph  cable.  The  con- 
ductor at  the  center  consists  of  a  number  of  strands  of  copper 
wire.  Surrounding  this  is  a  layer  of  gutta  percha,  and  the  whole 
is  protected  by  a  covering  of  tarred  hemp  and  steel  wire. 


2 

Fig.  31. 


The  conductor  and  metal  sheath  of  the  cable,  together  with 
the  intervening  insulator,  constitute  a  condenser  of  large  electro- 
static capacity.  The  effect  of  this  large  electrostatic  capacity  is 


MISCELLANEOUS    PRACTICAL  APPLICATIONS. 


329 


as  follows  :  At  the  instant  a  battery  is  connected  to  a  cable  a  very 
large  current  begins  to  flow  into  the  cable.  Most  of  this  current 
goes  to  charge  the  cable,  and,  as  the  cable  becomes  charged,  the 
entering  current  falls  off  in  value,  settling  finally  to  a  steady  value 
which  is  determined  by  the  resistance  of  a  cable.  The  ordinates 
of  curve  A  in  Fig.  3  I  show  the  successive  values  of  the  current 
which  enters  a  cable  from  a  battery.  At  the  distant  end  of 
the  cable  an  infinitesimal  current  begins  almost  at  the  instant 
the  battery  is  connected  at  the  sending  station,  and,  as  the  cable 
becomes  charged,  this  current  rises  in  value  until  it  reaches  a 
steady  value  very  nearly  equal  to  the  steady  value  of  the  enter- 
ing current.  The  curve  By  Fig.  31,  shows  the  growth  of  cur- 
rent at  the  distant  end  of  a  cable  when  a  battery  is  connected  to 
the  near  end.  When  the  battery  is  disconnected  the  current 
which  enters  the  cable  ceases  at  once,  and  the  current  at  the  dis- 
tant end  drops  slowly  to  zero  as  the  accumulated  charge  flows  out 
of  the  cable. 

Distortion  of  current  pulses  by  a  cable.  — The  curve  a,  Fig.  32, 
shows  the  character  of  the  current  pulse  which  enters  a  cable  when 


TIMB 


Fig.  32. 


a  battery  is  momentarily  connected  to  the  cable,  and  the  curve  b 
shows  the  character  of  the  current  pulse  which  flows  out  at  the 
distant  end  of  the  cable.  The  action  of  a  cable  in  thus  alter- 


330        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

ing  the  character  of  a  current  pulse  is  called  distortion.  Land 
lines  distort  current  pulses  to  some  extent,  and  the  distortion 
seriously  impairs  the  distinctness  of  telephonic  transmission  if  the 
land  line  is  fairly  long  (see  Art.^147). 

The  curve    a,    Fig.  33,   represents  four  short  current  pulses 


hhhh 


TIME 


Fig.  33. 

sent  into  a  cable  at  one  end,  and  the  curve  b  represents  the  re- 
sultant pulse  of  current  which  flows  out  of  the  cable  at  the  other 
end.  The  receiving  instrument  in  submarine  telegraphy  is  a  gal- 
vanometer  which  is  arranged  to  trace  the  resultant  current  curve  at 
the  receiving  end  of  the  cable,  and  the  separate  current  pulses  that 
are  sent  into  the  cable  at  the  sending  end  are  inferred  from  the 
slight  kinks  in  the  curve  which  is  traced  by  the  receiving  instrument. 
The  distortion  of  electric  current  pulses  by  a  submarine  cable 
is  analogous  to  the  distortion  of  pulses  of  water  current  by  a  long 
thin-walled  rubber  tube. 

35.  The  syphon  recorder  is  the  receiving  instrument  used  in 
submarine  telegraphy.  It  consists  of  a  D'Arsonval-type  gal- 
vanometer, the  moving  coil  of  which  is  attached  by  means  of  a 
fine  thread  to  a  syphon  of  very  fine  glass  tube.  This  syphon 
takes  ink  from  a  small  reservoir  and  traces  an  ink  line  upon  a 
moving  paper  ribbon.  When  the  galvanometer  coil  is  quiet  a 
straight  line  is  traced  upon  the  moving  paper.  When  signals  are 
being  received  the  varying  current  which  flows  through  the  gal- . 
vanometer  coil  causes  the  coil  to  move  and  the  glass  tube  traces 
a  wavy  line  upon  the  moving  paper.  It  is  necessary  for  the 


MISCELLANEOUS    PRACTICAL   APPLICATIONS. 


331 


syphon  to  move  sidewise  with  the  utmost  freedom,  and  therefore 
the  tip  of  the  syphon  cannot  be  allowed  to  rest  against  the  mov- 
ing paper.  This  difficulty  was  overcome  in  the  early  form  of  the 
syphon  recorder  *  by  keeping  the  ink  reservoir  and  syphon  highly 
charged  with  electricity  by  means  of  an  influence  machine,  thus 
causing  the  ink  to  issue  from  the  tip  of  the  syphon  in  the  form  of 
a  fine  jet.  In  the  present  form  of  the  recorder  the  syphon  is 


Fig.  34. 

kept  vibrating  rapidly  against  the  paper  so  as  to  trace  a  finely 
dotted  line  as  the  paper  moves  while  at  the  same  time  the  side- 
wise  motion  of  the  syphon  is  not  hindered  by  friction.  The  essen- 
tial features  of  the  syphon  recorder  are  shown  in  Fig.  34. 

*  The  syphon  recorder  was  devised  by  Lord  Kelvin,  who  contributed  more,  perhaps, 
to  the  development  of  transatlantic  telegraphy  than  any  other  man.  In  an  article  by 
Professor  W.  E.  Ayrton,  which  appeared  in  the  London  Times  shortly  after  Lord  Kel- 
vin's death  (reprinted  in  Popular  Science  Monthly  for  March,  1908),  much  interesting 
information  is  given  concerning  what  Kelvin  did  for  submarine  telegraphy.  "When 
signals  through  the  1858  Atlantic  cable  became  weak,  and  a  message  from  the  Presi- 
dent to  our  Queen  took  thirty  hours  in  transmission  although  containing  only  150 
words,  and  which  would  need  only  three  or  four  minutes  to  transmit  through  any  one 
of  our  good  Atlantic  cables  of  to-day,  the  only  remedy  of  those  who  looked  down 
upon  the  theories  of  the  young  Glasgow  professor  was  to  use  Whitehouse's  "thunder 
pump,"  a  magneto-electric  machine  which  produced  a  sudden  large  electromotive 
force  when  the  armature  of  the  permanent  magnet  was  jerked  off  the  poles  of  the 


33 2        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 

36.  The  telephone  consists  of  a  thin  sheet-iron  diaphragm  D, 
Fig.  35,  which  is  very  near  to  one  end  of  a  steel  magnet  M  with 
a  winding  of  fine  insulated  wire  C. 

The  action  of  the  telephone  as  a  transmitter.  — When  the  tele- 
phone first  came  into  use,  the  same  instrument  was  used  as  trans- 


Fig.  35. 

mitter  and  receiver,  being  moved  alternately  from  mouth  to  ear  of 
the  speaker.  The  action  of  a  telephone  as  a  transmitter  is  as 
follows  :  The  coil  C  being  near  the  en'd  of  the  magnet  M,  only  a 
portion  of  the  magnetic  flux  from  M  passes  through  the  coil. 
When  the  diaphragm  moves  nearer  to  the  end  of  the  magnet,  a 
greater  portion  of  the  magnetic  flux  from  the  magnet  passes  through 
C,  and  when  the  diaphragm  moves  farther  away  from  the  magnet, 
a  smaller  portion  of  the  magnetic  flux  from  the  magnet  passes 
through  C.  Thus,  as  the  diaphragm  D  vibrates,  the  magnetic 
flux  through  the  coil  C  increases  and  decreases.  This  pulsa- 
tion of  the  flux  through  the  coil  C  induces  an  electromotive 
force  in  the  coil,  and  this  induced  electromotive  force  produces  a 
current  in  the  coil  and  in  any  circuit  to  which  the  coil  is  con- 
nected. This  induced  current  flows  in  one  direction  while  the 
diaphragm  is  moving  towards  the  magnet,  and  in  the  other  direc- 
tion while  the  diaphragm  is  moving  away  from  the  magnet. 

The  action  of  a  telephone  as  a  receiver.  —  If  a  current  passes 
through   the  coil    C  first  in  one  direction  and  then  in  the  other 

magnet.  But  these  shocks  only  sent  sparks  through  the  gutta-percha  insulating  coat- 
ing and  hurried  the  poor  cable  to  its  doom,  so  that  even  the  three  words  per  minute 
which  would  have  been  the  utmost  limit  of  speed  possible  had  this  cable  been  entirely 
uninjured,  were  replaced  by  absolute  silence." 


MISCELLANEOUS    PRACTICAL   APPLICATIONS. 


333 


direction,  the  magnet  M  will  be  alternately  weakened  and 
strengthened,  the  force  with  which  the  magnet  attracts  the  dia- 
phragm will  vary  accordingly,  and  the  diaphragm  will  be  caused 
to  move  to  and  fro  in  unison  with  the  reversals  of  current. 

Consider  two  telephones,  A  and  B,  connected  in  circuit.  A 
sound  strikes  the  diaphragm  of  telephone  A  and  causes  the 
diaphragm  to  vibrate.  Telephone  A  acts  as  a  transmitter,  and 
telephone  B  acts  as  a  receiver,  as  explained  above,  and  the  dia- 
phragm of  telephone  B  is  caused  to  vibrate  in  a  manner  exactly 
similar  to  the  vibrations  of  the  diaphragm  of  telephone  A,  and 
thus  the  diaphragm  of  telephone  B  reproduces  the  original 
sound. 

37.  The  carbon  transmitter.  —  The  alternating  current  which  is 
produced  by  a  telephone  acting  as  a  transmitter  is  very  weak 
even  when  the  transmitter  telephone  is  exposed  to  a  loud  sound. 
The  carbon  transmitter  is  an  arrangement  by  means  of  which  a 
vibrating  diaphragm  may  control  a  strong  battery  current  and 


line 


line 


battery 

Fig.  36. 

cause  a  strong  induced  current  to  surge  back  and  forth  through 
the  telephone  line  in  unison  with  the  movements  of  the  diaphragm. 
The  arrangement  of  the  carbon  transmitter  is  shown  in  Fig.  36. 
The  current  from  a  battery  passes  through  the  primary  P  of 
a  small  induction  coil  and  through  a  mass  of  granular  carbon  C 
which  lies  between  a  carbon  block  B  and  a  diaphragm  DD. 
The  electrical  resistance  of  the  granular  carbon  varies  with  the 


334        ELEMENTS  OF  ELECTRICITY  AND    MAGNETISM. 

pressure  exerted  upon  it  by  the  vibrating  diaphragm,  this  causes 
the  battery  current  to  fluctuate,  the  fluctuating  battery  current 
induces  an  alternating  current  in  the  secondary  6"  of  the  induction 
coil  and  this  alternating  current  passes  over  the  line  and  actuates 
the  receiver  telephone  at  the  distant  station. 

38.  Wireless  telegraphy.  *  —  The  intensity  'of  the  magnetic 
field  in  the  neighborhood  of  an  isolated  magnet  pole  decreases 
as  the  square  of  the  distance  increases,  and  the  intensity  of  the 
magnetic  field  at  considerable  distances  from  a  complete  magnet 
(having  two  opposite  poles)  decreases  as  the  cube  of  the  distance 
increases.  The  energy  of  a  magnetic  field  is  proportional  to  the 
square  of  the  field  intensity  and  therefore  the  energy  of  the  mag- 
netic field  in  the  neighborhood  of  an  isolated  pole  decreases  as 
the  fourth  power  of  the  distance  increases,  and  the  energy  of  the 
magnetic  field  at  considerable  distances  from  a  complete  magnet 
decreases  as  the  sixth  power  of  the  distance  increases.  The 
same  laws  of  decrease  of  the  energy  apply  in  the  case  of  the 
electric  field  due  to  an  isolated  charge  and  in  the  case  of  the 
electric  field  due  to  a  doublet  consisting  of  two  opposite  charges 
near  together,  respectively.  In  the  case  of  wave  motion  of  any 
kind  which  spreads  out  uniformly  in  all  directions  from  a  source, 
the  energy  falls  off  as  the  square  of  the  distance  increases. 
Therefore  an  enormously  greater  amount  of  energy  can  be 
brought  into  action  at  great  distances  from  a  source  of  disturbance 
by  wave  motion  than  by  actions  which  produce  a  steady  distribu- 
tion of  field.  This  remarkable  property  of  wave  motion  is  illus- 
trated by  the  familiar  fact  that  an  audible  effect  may  be  produced 
upon  the  ear  of  a  distant  person  by  the  wave  disturbance  in  the 
air  which  is  produced  by  a  vibratory  motion,  whereas  an  ex- 
tremely violent  but  steady  circulation  of  air  produced,  for  ex- 
ample, by  a  powerful  fan-blower,  does  not  lead  to  any  perceptible 
energy  manifestations  at  moderately  great  distances  from  the 
blower.  In  consequence  of  the  very  great  energy  manifestations 

*  This  article  describes  the  simple  original  arrangement  which  is  due  to  Marconi, 


MISCELLANEOUS    PRACTICAL   APPLICATIONS. 


335 


at  a  distance  due  to  wave  motion  as  compared  with  the  extremely 
small  energy  manifestations  at  a  distance  due  to  steady  actions, 
it  may  be  said  that  the  only  feasible  method  *  of  signalling  at 
moderately  great  distances  is  by  means  of  wave  motion. 

The  use  of  the  vocal  organs  for  producing  sound  waves  and 
of  the  auditory  organs  for  perceiving  them  at  a  distance,  consti- 
tutes the  most  familiar  example  of  "  wireless  "  signalling.  The 
term  wireless  telegraphy  is  applied  particularly  to  the  use  of  an 
electric  oscillator  for  producing  electric  waves  and  an  electric 
resonator  or  detector  of  any  kind  for  perceiving  the  waves  at  a 


ground 

Fig.  37. 

distance.     This  system  of  electric  wave  signalling  is  due  to  Mar- 
coni. 

The  sending  apparatus  or  oscillator.  —  A  charged  body,  an 
expanse  of  metal,  is  suspended  in  the  air  so  as  to  be  thoroughly 
insulated  from  the  earth.  This  body  of  metal  is  usually  made 
of  many  strands  of  wire  WWt  Fig.  37,  which  are  supported  by 
guy  wires  GG  from  poles,  as  shown,  the  guy  wires  being 


*  Where  the  energy  is  not  transmitted  along  a  well-defined  path  like  a  wire,  or  a 
pipe,  or  a  string. 


336        ELEMENTS  OF  ELECTRICITY  AND  MAGNETISM. 


Fig.  38. 


provided  with  insulating  links  //.  The  body  of  metal  WWt 
which  is  separated  from  the  ground  by  a  short  air  gap  gy 
is  connected  to  one  terminal  of  a  high  voltage  induction  coil,  the 
other  terminal  of  which  is  connected  to  the  earth,  the  body  of 
metal  WW  is  charged  until  the  air  gap  g  breaks  down,  the 
discharge  which  takes  place  is  oscillatory  in  character  as  ex- 
plained in  Chapter  IX,  and  electric  waves  pass  out  in  all  direc- 
tions from  WW. 

The  receiving  antenna.  —  A  long  vertical  wire  is  suspended  by 
insulating  supports  at  the  receiving  station  and  connected  to  earth 

through  a  device  which  is  called  a 
detector.  The  passage  of  the  electric 
waves  causes  electric  charge  to  surge 
up  and  down  in  this  vertical  wire  or 
antenna,  and  the  weak  alternating 
current  thus  produced  actuates  the  detector  and  produces  the  sig- 
nal at  the  receiving  station.  The  detector  which  was  used  in  the 
earlier  days  of  wireless  te-  • 

legraphy  was  the  coherer 
of  Branly.  The  essential 
parts  of  this  coherer  are 
shown  in  Fig.  38.  It  con- 
sists of  two  short  brass 
rods  between  which  is  a 
loose  mass  of  metal  filings. 
This  coherer  is  connected  'B . 
across  the  air  gap  of  the  re- 
ceiving antenna  or  resona- 
tor as  shown  in  Fig.  39,  in 
which  5  is  an  ordinary 
telegraph  sounder.  An 
auxiliary  device,  not  shown 
in  the  figure,  is  used  to 
keep  the  metal  filings  vibrating  slightly.  Under  these  condi- 
tions the  filings  do  not  conduct  the  batteiy  current  to  any 


I 


receiving 
antena 


to  ground 


Fig.  39. 


MISCELLANEOUS   PRACTICAL  APPLICATIONS. 


337 


perceptible  extent.  When,  however,  electric  waves  act  upon  the 
antenna,  the  slight  current  which  is  produced  in  the  antenna  is 
forced  through  the  filings  and  produces  what  seems  to  be  a 
welding  together  of  the  particles  of  the  filings  at  the  points 
of  contact.  At  any  rate,  as  long  as  a  slight  amount  of  current 
is  forced  through  the  filings  from  the  antenna,  the  filings  form  a 
good  conducting  path  for  the  battery  current  and  the  sounder  is 
excited,  but,  the  moment  the  electric  waves  cease,  the  vibratory 
motion  of  the  metal  filings  causes  them  to  become  detached  from 
each  other  and  the  battery  current  ceases. 

Figure  40  shows  the  trend  of  the  electric  lines  of  force  in  the 
electric  waves    as   they  approach  the   receiving  antenna.     The 


wave 


wave 


receiving 
antena 


//'/////'/////////////// 
ground 


'  ft /'/////  f/r/  r 

ground 

Fig.  40. 


magnetic  lines  of  force  are  horizontal  and  perpendicular  to  the 
plane  of  the  paper. 

39.  Electric  lighting.  —  One  of  the  most  extended  applications 
of  the  electric  current  is  in  the  production  of  artificial  illumination. 
This  is  usually  accomplished  by  the  heating  to  incandescence  of  a 
high  resistance  portion  of  a  circuit,  by  the  electric  current.  The 
high  resistance  portion  of  the  circuit,  together  with  its  mounting,  is 
called  an  electric  lamp.  Two  types  of  electric  lamp  are  in  gen- 
23 


33^        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 

eral  use,  namely,  the  glow  lamp,  or  incandescent  lamp,  and  the 
arc  lamp. 

The  glow  lamp  consists  of  a  fine  filament  or  wire  of  highly  re- 
fractory material  which  is  enclosed^in  a  glass  bulb  from  which  the 
air  is  exhausted.  In  the  older  type  of  glow  lamp  the  filament  is 
made  of  charred  vegetable  material  upon  which  a  dense  deposit  of 
carbon  is  formed  by  heating  it  in  the  vapor  of  gasoline.  The 
heating  is  accomplished  by  sending  an  electric  current  through 
the  filament.  The  carbon-filament  glow  lamp  consumes  from 
three  to  four  watts  for  each  candle  of  light  emitted.  Thus,  a  1 6- 
candle  carbon  filament  lamp  consumes  about  55  watts. 

Recently  several  varieties  of  metal-filament  glow  lamps  have 
been  placed  on  the  market.  The  earliest  of  these  was  the 
osmium  lamp,  the  filament  of  which  is  made  of  metallic  osmium 
which  is  sufficiently  refractory  to  stand  a  temperature  high  enough 
to  emit  one  candle  of  light  with  a  consumption  of  about  I  y2 
watts.  The  scarcity  of  metallic  osmium,  however,  was  a  serious 
obstacle  in  the  way  of  extensive  use  of  the  osmium  lamp.  The 
next  metal  filament  lamp  to  be  placed  on  the  market  was  the 
tantalum  lamp,  in  which  the  filament  consists  of  a  wire  of  metallic 
tantalum.  In  the  tungsten  lamp,  which  is  now  coming  into  ex- 
tensive use  in  Europe  and  America,  the  filament  consists  of  me- 
tallic tungsten.  The  carbon  filament  lamp  consumes  from  3  to  4 
watts  for  each  candle  of  light  emitted,  the  osmium  lamp  con- 
sumes about  I  y2  watts  per  candle  of  light  emitted,  the  tantalum 
lamp  consumes  about  2  watts  per  candle,  and  the  tungsten  lamp 
consumes  about  I  ^  watts  per  candle.  The  greatest  difficulty 
with  the  metal  filament  lamps  is  that  the  filament  must  be  exces- 
sively fine  to  give  a  low  candle  power  lamp  with  the  standard  volt- 
ages now  in  use  for  lighting  purposes  (no  volts  and  220  volts), 
because  of  the  low  specific  resistance  of  metals  as  compared  with 
carbon.  This  difficulty  is  greatly  enhanced  in  the  case  of  the 
tungsten  lamp  by  the  excessive  brittleness  of  the  material. 

The  arc  lamp.  —  When  an  electric  arc  is  formed  between  carbon 
points  as  described  in  Chapter  VIII,  the  carbon  points  become 


MISCELLANEOUS    PRACTICAL   APPLICATIONS. 


339 


Fig.  41. 


intensely  heated  and  give  off  a  very  brilliant  light.  The  arc 
lamp  is  a  mechanism  for  automatically  moving  two  carbon  rods 
so  that  a  steady  electric  arc  may  be  maintained  between  the  ends 
of  the  rod.  There  is  a  great  variety  of  arc  lamp  mechanisms  but 
the  following  description  will  serve  to  give  an  idea  of  their  action  : 
The  current  comes  into  the  lamp  and  divides  as  shown  in  Fig.  41. 
A  very  small  portion  of  the 
current  flows  through  a 
shunt  coil  B  without  pass- 
ing through  the  arc,  and  the 
remainder  flows  through 
the  coil  A  and  thence 
through  the  arc.  An  iron 
rod  AB,  passing  loosely 
into  the  two  coils  A  and 
B,  is  carried  upon  one  end 
of  a  lever  which  is  pivoted 
at  the  point  /.  The  other 
end  of  this  lever  is  provided  with  a  clutch  c  through  which  a 
smooth  brass  rod  bb  passes.  This  clutch  c  is  so  constructed 
that  it  releases  the  rod  bb  when  the  iron  rod  AB  is  raised,  thus 
allowing  the  carbons  to  come  together.  Each  of  the  coils  A 
and  B  acts  to  pull  the  rod  AB  into  itself,  and  a  spring  which 
is  attached  to  the  lever  is  so  adjusted  that  when  the  arc  is  burning 
properly  the  combined  action  of  this  spring  and  the  two  coils  A 
and  B  holds  the  lever  in  such  a  position  that  the  clutch  grips 
the  brass  rod  bb.  As  the  arc  continues  to  burn,  the  carbons 
are  slowly  consumed,  causing  the  gap  between  the  carbon  tips  to 
widen.  This  increases  the  resistance  of  the  arc  and  causes  a 
greater  portion  of  the  current  to  flow  through  the  shunt  coil  B 
which  pulls  up  on  the  iron  rod  AB,  moves  the  lever,  releases 
the  clutch,  and  allows  the  rod  bb  to  fall  slightly,  thus  bringing 
the  carbons  again  to  the  proper  position. 

A  variety  of  arc  lamps  have  been  developed  in  which  the  light 
is  emitted  by  the  arc  itself.     Thus  we  have  the  so-called  flaming- 


340        ELEMENTS  OF  ELECTRICITY  AND   MAGNETISM. 


arc  lamp  in  which  the  carbon  rods  are  impregnated  with  metallic 
salts,  the  vapors  of  which  give  an  intensely  luminous  arc.  An- 
other form  of  arc  lamp  in  which  the  arc  itself  is  intensely  luminous 
is  the  magnetite  arc  lamp  in  which  the  arc  is  formed  between  a 
rod  of  compressed  titanium  carbide  and  iron  oxide  (the  cathode) 
and  a  rod  of  copper  (the  anode).  The  result  is  the  vaporization 
of  the  iron  oxide  and  the  production  of  an  intensely  luminous 
arc. 

40.  The  electrolytic  interrupter  (Wehnelt).  —  The  primary  cir- 
cuit of  an  induction  coil  is  usually  interrupted  by  a  vibrating  reed 
or  spring  which  makes  and  breaks  contact  between  two  platinum 
points.  Wehnelt  discovered  that  the  sudden  generation  of 
oxygen  on  a  small  platinum  anode  in  dilute  sulphuric  acid  causes 
an  abrupt  stoppage  of  the  electric  current.  This  effect  is  utilized 
in  the  electrolytic  interrupter  as  follows  :  A  glass  jar  CC,  Fig. 
42,  is  rilled  with  dilute  sulphuric  acid  and  provided  with  two 


ELECTRIC  MAfN 


ELEOTKIC  MAIN 


PRIMARY  OF 
WnUCTIQN  COlt 


Fig.  42. 

electrodes  /  and  /.  The  anode  p  is  a  tip  of  platinum  wire 
projecting  from  a  glass  tube,  and  the  cathode  /  is  a  large  plate 
of  lead.  The  electromotive  force  between  the  mains,  which  must 
be  30  volts  or  more,  causes  a  sudden  rush  of  current  through  the 
cell  CC  and  through  the  primary  of  an  induction  coil.  This 
rush  of  current  generates  a  layer  of  oxygen  over  the  platinum 


MISCELLANEOUS   PRACTICAL  APPLICATIONS. 


341 


tip  which  stops  the  current  abruptly.  The  layer  of  oxygen  then 
collects  as  a  bubble  and  rises,  leaving  the  platinum  tip  again  in 
contact  with  the  acid  when  another  rush  of  current  takes  place, 
and  so  on.  From  200  to  1,500  interruptions  per  second  maybe 
produced  by  this  arrangement  according  to  the  size  of  the  plati- 
num tip,  the  inductance  of  the  circuit  and  the  value  of  the  electro- 
motive force. 

41.  Electric  welding.  Thomson's  process.  —  The  two  metal 
rods  to  be  welded  are  connected  to  the  terminals  of  an  electric 
generator  and  brought  into  contact  with  each  other.  The  cur- 
rent, flowing  across  the 
relatively  high  resistance 
contact,  heats  the  ends  of 
the  rods  to  the  melting 
temperature,  the  rods  are 
then  pushed  slightly  to- 
gether and  the  weld  is 
complete.  Alternating  cur- 
rent is  generally  used  in 
this  welding  process ;  a 
transformer  takes  current 
at  high  voltage  from  ordinary  supply  mains  and  delivers  a  very 
large  current  at  very  low  voltage  to  the  rods  to  be  welded. 

The  wet  process.  —  When  a  direct-current  generator  having  an 
electromotive  force  of  from  200  to  500  volts  is  connected  to  an 
electrolytic  cell  with  small  cathode,  the  cathode  becomes  intensely 
heated.  This  effect  is  utilized  for  welding  as  follows  :  The  two 
rods  a  and  by  Fig.  43,  which  are  to  be  welded  are  connected 
to  the  negative  terminal  of  the  dynamo  D.  The  positive  ter- 
minal of  the  dynamo  is  connected  to  a  metal  nozzle  from  which 
a  jet  of  salt  water  issues.  This  jet  impinges  upon  the  ends  of 
the  two  rods  and  quickly  fuses  them  together. 


Fig.  43. 


APPENDIX    E. 


MECHANICAL  AND  ELECTRICAL  ANALOGIES. 

The  mechanical  analogies  which  are  pointed  out  in  Art.  62  of 
Chapter  V,  in  Chapter  VI,  in  Arts.  89  and  93  of  Chapter  VII, 
and  in  Arts.  106,  107  and  108  of  Chapter  VI II  'are  here  collected 
together  for  convenience  of  reference,  and  the  mechanical  anal- 
ogies of  electrical  oscillations  are  added  : 

X=Vt  (I) 

in  which  x  is  the  distance 
traveled  in  t  seconds  by  a 
body  moving  at  velocity  v. 


in  which  W  is  the  work 
done  by  a  force  F  in  pull- 
ing a  body  through  the  dis- 
tance x. 

P=Fv          (7) 

in  which  P  is  the  power 
developed  by  a  force  F  act- 
ing upon  a  body  moving  at 
velocity  v. 

W=\mv*       (10) 

in  which  W  is  the  kinetic 
energy  of  a  mass  m  mov- 
ing at  velocity  v. 

F—mdV 

m  in 

in  which  F  is  the  force  re- 
quired to  cause  the  velocity 
of  a  body  of  mass  m  to  in- 

dv 

crease  at  the  rate    — . 
dt 


x=-.aF        (16) 
~-~  C9) 


t  =  ut                 (2) 

9=**            (3) 

in  which    0    is  the   angle 

in  which    q   is  the  electric 

turned  in    t  seconds  by  a 

charge  which  in   t   seconds 

body    turning   at    angular 

flows  through  a  circuit  car- 

velocity   u. 

rying  a  current   i. 

W=7)          (5) 

W=Eq           (6) 

in  which    W  is  the  work 

in  which     W  is  the  work 

done  by  a  torque  T  in  turn- 

done   by    an   electromotive 

ing     a    body   through   the 

force  E  in  pushing  a  charge 

angle  <j>. 

q   through  a  circuit. 

P=Tu           (8) 

P  =  Ei            (9) 

in  which    P  is  the   power 

in  which    P  is  the  power 

developed  by   a  torque    T 

developed    by    an    electro- 

acting on  a  body  turning  at 

motive  force  E  in  pushing  a 

angular  velocity   u. 

current  i  through  a  circuit. 

W=\Ktf        (n) 

W=\LP       (12) 

in  which    W  is  the  kinetic 

in  which    W  is  the  kinetic 

energy  of  a  wheel  of  mo- 

energy of  a  coil  of  induc- 

ment of  inertia    fC  turning 

tance  L  carrying  a  current  i. 

at  angular  velocity    u. 

E  =  L  —         (  *  5  ) 

in  which     T  is  the  torque 

in  which  E  is  the  electro- 

required to  cause  the  angu- 

motive   force     required   to 

lar  velocity  of  a  wheel   of 

cause  a  current  in  a  coil  of 

moment  of    inertia    K   to 

inductance  L  to  increase  at 

increase  at  the  rate    —  . 
dl 

di 
the  rate    -7. 
at 

^=bT        (17) 

(  <?O  ^ 

f=C£         (18) 

47T2Z               I 

T*       -         b           (2°} 

_2                  £•           V21) 

342 


MECHANICAL  AND   ELECTRICAL  ANALOGIES. 


343 


Fig.  a. 

A  body  of  mass  m  is  sup- 
ported by  a  flat  spring  S, 
clamped  in  a  vise  as  shown 
in  Fig.  a.  A  force  F  push- 
ing sidewise  on  m  moves  it 
a  distance  x,  which  is  pro- 
portional to  F,  according 
to  equation  (16).  When 
started  the  body  m  will  con- 
tinue to  vibrate  back  and 
forth  and  the  period  r  of  its 
vibrations  is  determined  by 
equation  (19). 


wire 


body 


Fig.  b. 

A  body  of  moment  of  in- 
ertia K  is  hung  by  a  wire 
as  shown  in  Fig.  b.  A 
torque  T  acting  on  the  body 
will  turn  the  body  and  twist 
the  wire  through  an  angle 
(j>,  which  is  proportional  to 
T,  according  to  equation 
(17).  When  started,  the 
body  will  vibrate  about  the 
wire  as  an  axis  and  the 
period  r  of  its  vibrations 
is  determined  by  equation 
(20). 


Fig.  c. 

A  condenser  C  is  con- 
nected to  the  terminals  of 
a  coil  of  inductance  L  as 
shown  in  Fig.  c.  An  elec- 
tromotive force  E  acting 
anywhere  in  the  circuit 
pushes  into  the  condenser 
a  charge  q,  which  is  pro- 
portional to  J5,  according 
to  equation  (18).  When 
started  the  electric  charge 
will  surge  back  and  forth 
through  the  coil,  constitut- 
ing what  is  called  an  oscil- 
latory current  and  the  period 
of  one  oscillation  is  deter- 
mined by  equation  (21 ). 


INDEX. 


Abampere,  definition  of,  98 
Abcoulomb,  definition  of,  161 
Ab farad,  definition  of,  166 
Abhenry,  definition  of,  143 
Abohm,  definition  of,  99 
Absolute  electrometer,  the,  183 

measurements,  276 

units,  98 

Abvolt,  definition  of,  99 
Aging  of  permanent  magnets,  83 
Alloys,  resistivities  of,  28 
Alternating  current,  125 
Alternating-current  dynamo,  the,  125 

transformer,    mechanical     ana- 
logue of,  195 
the,  133 

electromotive  force,  125 
Alternator,    see    alternating-current  dy- 
namo 

Aluminum,  manufacture  of,  5 
Ammeters  and  voltmeters,  42 
Ammeter,  the,  2 

shunts,  49 
Ampere,  definition  of,  98 

the  international  standard,  8 
Analogies,  mechanical  and  electrical,  342 
Anion,  definition  of,  6,  10 
Anode,  definition  of,  5 
Arc,  the  electric,  232 

lamp,  the  flaming,  340 
the  magnetite,  340 
Armature  of  direct-current  dynamo,  128 

of  the  alternator,  126 
Astatic  system  of  magnets,  112 
Atmospheric  electricity,  321 
Attraction,  electrostatic,  163 

Ballistic  galvanometer,  the,  161,  285 
Battery,  the  electric,  see  voltaic  cell 
the  storage,  see  storage  cell 


Bell  telephone,  the,  322 
Bismuth  inductometer,  the,  290 
Blow-out,  the  magnetic,  152 
Branched  circuits,  44 
Branly's  coherer,  336 
Brush  discharge,  the,  216 

Calcium  carbide,  manufacture  of,  26 
Canal  rays,  227 
Capacity,  electrostatic,  165 
measurement  of,  165,  286 
mechanical  analogue  of,  168 
units  of,  1 66 

Carbon  transmitter,  the,  333 
Carhart,  H.  S.,  Determination  of  Elec- 
trochemical Equivalent  of  Silver ; 
8 

On  the  Thermodynamics  of  the  Vol- 
taic Cell,  35 

and  Patterson,  Electrical  Measure- 
ments, 162 

Cathion,  definition  of,  6,  10 
Cathode,  definition  of,  5 

rays,  227 
Centimeter,  the,  as  a  unit  of  inductance, 

H3 
Charge,  concentrated,  electric   field  due 

to,  179 

electric,  see  electric  charge 
Charges,  concentrated,  attraction  and  re- 
pulsion of,  1 80 
Charging  by  contact  and  separation,  199 

by  influence,  202 
Chemical  effect  of  the  electric  current,  I, 

4,5 

Choke  coil,  the,  151 
Chromic-acid  cell,   14 
Circular  mil,  definition  of,  27 
Clarke  standard  cell,  1 6,  284 
"Climax"  metal,  28,  30 


345 


346 


INDEX. 


Coherer,  the  Branly,  336 

Collector  rings  of  the  alternator,  126 

Combined    resistance    of    a   number    of 

branches  of  a  circuit,  47 
Commutator  of  direct-current  dynamo,  129 
Compass  correctors,  adjustment  of,  310 
quadrantal,  304,  306 
semicircular,  301 

errors,  300 

heeling  corrector  of,  310 

the,  62,  293 
Concentrated  charge,  field  due  to,  179 

charges,  attraction  and  repulsion  of, 

1 80 
Condenser,  mechanical  analogue  of,  1 66 

potential  energy  of  a  charged,  170 

the,  1 60,  165 

Conductivity,  definition  of,  27 
Conductors  of  electricity,  I 
Convective  discharge,  214 
Copper,  electrolytic  refining  of,  5 
Coulomb,  definition  of,  161 
Coulomb's  Law  of  magnetic  attraction,  64 
Coulombmeter,  the,  8 

the  silver,  21 
Crookes  tube,  the,  226 
Current  density  at  an  electrode,  8 

measurement  of,  276 
Cycle,  definition  of  the,  127 

Danneel's  Electrochemistry -,  5 
Daniell  cell,  see  gravity  cell 
D' Arson val  galvanometer,  the,  113 
Decay  of  current  in  an  inductive  circuit, 

149 

Declination,  magnetic,  293 
Demagnetization  by  reversals,  84 
Diamagnetic  substances,  87 
Dielectric,  inductivity  of,  1 68 

strength,  174 

strengths,  table  of,  1 76 

the,  163 
Dip,  magnetic,  293 

needle,  the,  293 
Diplex  telegraphy,  325 
Direct-current  dynamo,  fundamental  equa- 
tion of,  131 


Direct-current  dynamo,  the,  127 
Discharge  by  convection,  214 
by  disruption,  214 
from  metallic  points,  218 
Djsruptive  discharge,  214 
Dissociation  theory  of  electrolysis,  appli- 
cations of,  1 2 
of  electrolysis,  the,  10 
Dolezalelc,  The  Theory  of  the  Lead  Stor- 
age Cell,  1 8 

Doubler,  the  electrical,  196 
Drop  of  voltage,  39,  40 
Dry  cell,  the,  13 
Dubois,  H. ,  Magnetic  Circuit  in  Theory 

and  Practice,  81 
Duplex  telegraphy,  326 
Dynamo,      direct-current,     "fundamental 

equation  of,  131 
the  alternating-current,  125 
the  direct-current,  127 
the,  117,  124 
Dynamos,  types  of,  124 

Edison-LaLande  cell,  15 
Eddy  currents,  136 

examples  of,  136,  137 
Electric  absorption,  166 
arc,  the,  232 
charge,  160 

measurement  of,  161 
units  of,  161 

current,  chemical  effect  of,  I,  4,  5 
direction  of,  6 
heating  effect  of,  4,  25 
hydraulic  analogue  of,  4 
magnetic  effect  of,  I,  93 
measurement  of,  by  electrolysis, 

7 

strength    of,    magnetically   de- 
fined, 98 

discharge  in  gases,  224 
field,  direction  of,  174 

due  to  a  concentrated    charge, 

179 

energy  and  tension  of,  185 
intensity  of,  172 
mechanical  analogue  of,  164 


INDEX. 


347 


Electric  field,  mechanical  conception  of, 

242 
the,  163 

flux,  177 

furnace,  the,  26 

generator,  the,  124 

lighting,  337 

machine,  the  frictional,  201 

machines  of  the  influence  type,  203 

momentum,  141 

definition  of,  155 

motor,  the,  124 

oscillations,  242 

oscillator,  252,  254 

osmosis,  321 

potential,  1 86 

spark  in  a  gas,  224 
the,  164,  215 

wave  distortion,  269 

waves,  242 

welding,  341 

whirl,  the,  219 
Electrical  and  mechanical  analogies,  342 

conductors,  I 

doubler,  the,  196 

insulators,  I 

measurements,  276 

resistance,  25 

Electrically  charged  bodies,  162 
Electrochemical  equivalent, definition  of,9 
Electrodes,  definition  of,  5 
Electrodynamometer,  the,  109 

Siemens',  no 

the  Weber,  1 10 
Electrokinetic  energy,  141 
Electrolysis,  5 

current  density  in,  8 

dissociation  theory  of,  10 

Faraday's  Laws  of,  9 
Electrolyte,  definition  of,  5 
Electrolytic  cell,  definition  of,  5 
Electromagnet,  the,  61 
Electromagnetic  system  of  units,  1 80 

theory  a  branch  of  mechanics,  117 

wave,  velocity  of,  267 
Electromagnetism  and  ferromagnetism,  6 1 
Electromechanics,  219 


Electrometer,  the  absolute,  183 

the  quadrant,  see  electrostatic  volt- 
meter 

Electromotive  force,  definition  of,  35 
drop,  39,  40 

hydraulic  analogue  of,  36 
induced,  117 
measurement  of,  282 
Electrons,  222 
Electrophorus,  the,  202 
Electroplating,  4 
Electroscopes,  209 
Electrostatic  attraction,  163 

of  concentrated  charges,  180 
of  parallel  plates,  181 
capacity,  see  capacity 
system  of  units,  1 80 
voltmeter,  the,  184 
Electrostatics,  the  phenomena  of,  194 
Energy,  potential,  of  a  charged  condenser, 

170 

stream  in  electromagnetic  field,  245 
Esty,    William,    Elements  of  Electrical 

Engineering,  8 1 
Ewing,  J.   A.,    Magnetic   Induction   in 

Iron  and  other  Metals,  8 1 
Ewing' s  theory  of  the  magnetization  of 
iron,  86 

Farad,  definition  of,  166 

Faraday  units,  see  electrostatic  system  of 

units 
Faraday's  experiment,  212 

discovery  of    induced  electromotive 
force,  1 20 

Law's  of  electrolysis,  9 
Ferromagnetism    and    electromagnetism, 

61 
Field,  electric,  see  electric  field 

magnetic,  see  magnetic  field 

windings,  shunt  and  series,  129 
Flaming-arc  lamp,  340 
Flinders'  bar,  307 
Fluoroscope,  the,  230 
Flux,  electric,  177 
Flux-turns,  definition  of,  155 
Focusing  tube,  the,  231 


348 


INDEX. 


Franklin,  E.  C,  Application  of  Dissocia- 
tion Theory,  12 

Franklin,  W.  S. ,  Elements  of  Electrical 
Engineering,  81 

Frequency,  definition  of,  127 

Galvanic  cell,  see  voltaic  cell 
Galvanometer  shunts,  49 
the  ballistic,  161 
the  D' Arson val,   113 
the  Kelvin,  1 1 2 
the  tangent,  104 

Gaussin's  error  of  the  compass,  310 
Gauss's  method  for  measuring  the  hori- 
zontal component  of  the  earth's  mag- 
netic field,  74,  287 
Geissler  tube,  the,  226 
Generator,  the  electric,  124 
Gibbs,  H.  D.,  Application  of  Dissocia- 
tion Iheory,  12 
Glow  lamp,  the,  339 
Gold  leaf  electroscope,  209 
Gravity  cell,  the,  14 
Gray,  Andrew,  Absolute  Measurements, 

144 
Treatise    on    Magnetism    and 

Electricity,  292 

Grenet  cell,  see  chromic-acid  cell 
Growth  of  current  in  an  inductive  circuit, 
146 

Hadley,   H.  E.,  Magnetism   and  Elec- 
tricity for  Students,  3 1 6 
Hall  effect,  the,  318 
Heating  effect  of  electric  current,  4,  25 
Heaviside,  Electromagnetic  Theory,  271 
Heeling  corrector  of  compass,  310 

error  of  compass,  309 
Helmholtz's  Theory  of  Monocyclic  Sys- 
tems, 117 

Henry,  definition  of  the,  143 
Hertz,  On  Electric  Waves,  264 
Hydraulic  analogue  of  electromotive  force, 

36 
of  the  electric  current,  4 

Inclination,  magnetic,  293 


Induced  electromotive  force,  117 

law  of,  121,  123 
Inductance,  141 

definition  of,  142 
/«•       measurement  of,  144 

mechanical  analogue  of,  144 

mutual,  definition  of,  156 

of  a  long  solenoid,  154 

units  of,  143  '" 
Induction  coil,  the,  131 
Inductive   circuit,  growth  and   decay  of 

current  in,  146,  149 
Inductivity  of  a  dielectric,  168 
Inductivities,  table  of,  169 
Inductometer,  the  bismuth,  290 
Insulation    resistance,    measurement   of, 

282 

Insulators  of  electricity,  I 
Intensity  of  electric  field,  172 

of  magnetic  field,  66 

of  magnetization,  84 
International  standards,  history  of,  by  F. 

A.  Wolff,  276 
Ions  in  gases,  222 
lonization  of  a  gas,  223 
Iron,  magnetization  of,  8 1 

Jones's  Theory  of  Electrolytic  Dissocia- 
tion, 5 
Joule's  Law,  25 

application  of,  to  a  portion  of  a 
circuit,  38 

Kelvin  galvanometer,  the,  1 12 
Kerr  effect,  the,  318 
Key,  the  telegraph,  324 

Lamination,  136 

Lamp,  the  electric,  338 

Larmor,  Joseph,  &ther  and  Matter,  242 

Leblanc's  Electrochemistry,  5 

Lenz's  Law,  117 

examples  of,  I2O,  136 
Lighting,  electric,  337 
Lightning  arrester,  the,  152 

protection,  322 
Line  of  force,  definition  of,  65,  70 


INDEX. 


349 


Lippmann's  electrometer,  320 
Local  action  and  voltaic  action,  1 6 
Lodge,  Sir  Oliver,  Electrons,  236, 

Lightning      Conductors      ana 

Lightning  Guards,  322 
Modern    Views  of  Electricity, 

242 
Lorenz's  method  for  measuring  resistance, 

278 

Lyndon,  Storage  Battery  Engineering, 
18 

Magnet,  behavior  of,  in  a  uniform  field, 

73 

in  a  non-uniform  field,  74 
near  an  electric  wire,  94 
the,  6 1 
pole,  algebraic  sign  of,  65 

and   flux,  general   relation  be- 
tween, 71 
strength  of,  63 
poles,  62 

distributed  and  concentrated,  63 
the  permanent,  62 
Magnets,  astatic  system  of,  1 12 

permanent,  83 

Magnetic  attraction,  Coulomb's  Law,  64 
blow-out,  the,  152 
effect  of  the  electric  current,  I,  93 
elements,  293 
field,  action  of  upon  suspended  coil, 

1 06 

around  a  magnet  pole,  67 
inside  of  a  long  solenoid,  102 
intensity  of,  66 
measurement  of,  287 
mechanical  conception  of,  242 
near  a  long  slim  pole,  71 
the,  65 

tension  and  energy  of,  76 
uniform,    action    of,    upon    a 

magnet,  73 
and  non-uniform,  67 
non-uniform,  action  of,  upon  a 

magnet,  74 

fields,  composition  of,  68 
resolution  of,  69 


Magnetic  figures,  65 
flux  and  pole  strength,  general  re- 
lation between,  71 
definition  of,  69 
measurement  of,  287 
from  a  magnet  pole,  70 
maps,  294 

rotation  of  polarization  of  light,  318 
saturation,  84 
separator,  the,  76 
Magnetism  of  iron,  6 1 
residual,  83 
terrestrial,  292 
Magnetite  arc  lamp,  340 
Magnetization,  intensity  of,  84 
of  iron,  8 1 

Ewing's  theory  of,  86 
molecular  theory  of,  85 
Manganin,  32 

Marconi,  wireless  telegraphy,  334 
Maxwell,  definition  of  the,  70 
Maxwell's  Electricity  and  Magnetism, 

162 

Measurement  of  current,  276 
of  capacity,  286 

of  electric  current  by  electrolysis,  7 
of  electromotive  force,  282 
of  insulation  resistance,  282 
of  magnetic  fields,  287 

flux,  287 
of  power,  284 
of  resistance,  26,  278 
Measurements,  electrical,  276 
Mechanical    analogies    of  electromotive 

force  and  resistance,  117 
of  induced  electromotive  force, 

118 
analogue  of  condenser,  166 

of    electrically  charged  bodies 

and  of  the  electric  field,  164 
of  inductance,  144 
and  electrical  analogies,  342 
conception  of  electric  field,  242 

of  magnetic  field,  242 
theory  versus  atomic  theory  of  elec- 
tricity, 219 
Microfarad,  definition  of,  1 66 


350 


INDEX. 


Mil,  definition  of  the,  27 

Millivoltmeter,  the,  50 

Momentum,  electric,  141 

Morse  telegraph,  the,  3,  323 

Motor,  the  electric,  1 24 

Multiplying  coils  for  voltmeters,  50 

Multipolar   dynamo,   the   direct-current, 

130 
Mutual  inductance,  definition  of,  156 

Napier's  diagram,  312 
Non-inductive  circuits,  143 

Ohm,  definition  of,  26,  99 

the  international  standard,  26 
Ohm's  Law,  37 

application  of,  to  a  portion  of  a 

circuit,  38 
Open-circuit  cells  and  closed-circuit  cells, 

20 

Oscillations,  electric,  242 
Oscillator,  the  electric,  252,  254 
Osmosis,  electric,  321 
Ozone,  the  production  of,  233 

Parallel  and  series  connections,  44 

Paramagnetic  substances,  87 

Patterson  and  Carhart,  Electrical  Meas- 
urements >  162 

Patterson,  G.  W. ,  Determination  of  Elec- 
trochemical, Equivalent  of  Silver,  8 

Peltier  effect,  the,  316 

Permanent  magnet,  the,  62,  83 
magnets,  aging  of,  83 

Piezo-electricty,  318 

Pith-ball  electroscope,  the,  209 

Polarized  relay,  the,  324 

Polarization  of  the  voltaic  cell,  43 

Poles  of  a  magnet,  62 

Potential-difference,  definition  of,  40 

Potential,  electric,   1 86 

energy  of  a  charged  condenser,  170 

Potentiometer,  the,  282 

Power,  measurement  of,  284 

Poynting,  J.  H.,  On  the  Energy  Stream, 

245 
Primary  coil  of  induction  coil,  132 


Printing  telegraph,  the,  327 

Pyro-electricity,  317 

Pyrometer,  the  thermo-electric,  315 

Quadrant  electrometer,  the,  see  electro- 
static voltmeter 

Quadrantal  compass  correctors,  304,  306 
Quadruplex  telegraphy,  326 

Radio-activity,  234 
Relay,  the  polarized,  324 

the  telegraph,  323 
Residual  magnetism,  83 
Resistance,    combined,  of   a  number   of 
branches,  47 

electrical,  25 

measurement  of,  26,  278 

specific,  see  resistivity 

temperature  coefficient  of,  33 
coefficients  of,  table  of,  28 

variation  of  with  temperature,  31 
Resistivity,  definition  of,  27 
Resistivities  of  alloys,  28 

table  of,  28 
Rheostat,  the,  30 

the  water,  30 
Roentgen  rays,  230 
Rosa,  E.  B.,  papers  on  Measurement  of 

Inductance,  144 
Ruhmkorff  coil,  the,  131 
Rutherford,   E.,  Radio-activity  and  Ra- 
dio-active Transformations,  234 

Saturation,  magnetic,  84 
Secondary  coil  of  induction  coil,  132 
Selenium,  properties  of,  321 
Self-induced  electromotive  force,  146 
Self-induction,  coefficient  of,  see   induc- 
tance 

Semicircular  compass  correctors,  301 
Series  and  parallel  connections,  44 

and  shunt  field  windings,  129 

dynamo,  the,  129 
Ship's  compass,  the,  298 

magnetism,  the,  299 
Shunt  and  series  field  windings,  129 

dynamo,  the,  129 


INDEX. 


351 


Shunts,  use  of,  with  galvanometers  and 

ammeters,  49 

Siemens'  electrodynamometer,  no 
Silver  coulombmeter,  the,  21 
Soddy,  Frederick,  Radio-activity,  234 
Solenoid,  magnetic  field  inside  of,  102 
Sounder,  the  telegraph,  323 
Spark  at  break,  141 

gauge,  the,  177 

micrometer,  the,  177 

electric,  in  a  gas,  224 

the  electric,  164 

Sparking  distances  in  air,  table  of,  177 
Specific  inductive  capacity,  see  inductivity 
Standard  cell,  Clarke,  16 

cells,  284 

ohm,  the  international,  26 
Step-up  and  step-down  transformation,  134 
Storage  cell,  the,  18 
Strength,  dielectric,  174 

of  a  magnet  pole,  63 

of  current,  magnetically  defined,  98 
Strengths,  dielectric,  table  of,  176 
Submarine  telegraphy,  328 
"Superior"  metal,  28,  30 
Syphon  recorder,  the,  330 

Table  of  dielectric  strengths,  176 

of  inductivities  of  various  dielectrics, 
169 

of  sparking  distances  in  air,  177 
Tangent  galvanometer,  104 
Temperature  coefficients  of  resistance,  33 

coefficients  of  resistance,  table  of,  28 
Telegraph,  the  Morse,  3,  323 

the  printing,  327 
Telegraphy,  diplex,  325 

duplex,  326 

quadruplex,  326 

submarine,  328 

wireless,  334 
Telephone,  the,  322 
Terrestrial  magnetism,  292 
Thermo-electricity,  315 
Thermo-element,  315 
Thermopile,  the,  315 
Thomson  effect,  the,  317 


Thomson,  J.  J.,  Applications  of  Dynamics 
to  Physics  and  Chemistry,  117 

Thomson's,  J.  J.,  Conduction  of  Elec- 
tricity Through  Gases,  225 

Toepler-Holtz  machine,  the,  204 

Transformation,  step-up  and  step-down, 

J34 

Transformer,  current   and   electromotive 

force  relations  of,  135 
the  alternating-current,  133 
the   alternating  current,  mechanical 

analogue  of,  195 
Transmitter,  the  carbon,  333 
the  telephone,  333 

Units,  electromagnetic  system  of,  180 
electrostatic  system  of,   180 

Vacuum  tube,  the,  226 
Volt,  definition  of,  36,  99 

definition  of,  on  the  basis  of  Ohm's 

Law,  38 

Voltaic  action  and  local  action,  16 
cell,  the,  12 

cells  for  closed  circuits,  20 
for  open  circuits,  20 
types  of,  13 
Voltage  drop,  39,  40 
Voltameter,  the,  see  the  coulombmeter 
Voltmeters  and  ammeters,  42 
Voltmeter  multiplying  coils,  50 
the  electrostatic,  184 

Water  waves  in  a  canal,  257 
Wattmeter,  the,  284 
Wave  distortion,  261,  269 

electromagnetic,  velocity  of,  267 

the  electromagnetic,  262 
Waves,  electric,  see  electric  waves 

in  a  canal,  257 

Weber's  electrodynamometer,  no 
Wehnelt's  interrupter,  340 
Welding,  electric,  341 
Weston  cell,  the,  284 
Wheatstone'  s  bridge,  282 
Wimshurst  machine,  the,  206 
Wireless  telegraphy,  334 

Z*eman  effect,  the,  319 


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